<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Implicit Differentiation: x² + y² = 25</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }
    
    body {
      background: #ffffff;
      color: #0f172a;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
      max-width: 800px;
      margin: 0 auto;
    }
    
    .nav-bar {
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
      flex-wrap: wrap;
    }

    .step-pills {
      display: flex;
      gap: 6px;
      flex-wrap: wrap;
    }

    .pill {
      background: #f8fafc;
      border: 1px solid #94a3b8;
      border-radius: 999px;
      padding: 4px 10px;
      font-size: 12px;
      font-weight: 600;
      color: #334155;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 5px;
      transition: all 0.3s ease;
    }

    .pill .num {
      background: #e2e8f0;
      color: #1e293b;
      width: 16px;
      height: 16px;
      border-radius: 50%;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      font-size: 10px;
      font-weight: 700;
      transition: all 0.3s ease;
    }

    .pill:hover {
      background: #e2e8f0;
      border-color: #64748b;
    }

    .pill.active {
      background: #0284c7;
      color: #ffffff;
      border-color: #0284c7;
      box-shadow: 0 1px 3px rgba(2, 132, 199, 0.3);
    }

    .pill.active .num {
      background: #ffffff;
      color: #0284c7;
    }

    .ctrl-btn {
      background: #ffffff;
      border: 1px solid #94a3b8;
      border-radius: 6px;
      padding: 4px 10px;
      font-size: 12px;
      font-weight: 600;
      cursor: pointer;
      color: #0f172a;
      transition: all 0.3s ease;
    }

    .ctrl-btn:hover {
      background: #f1f5f9;
    }

    .main-grid {
      display: grid;
      grid-template-columns: 1.15fr 0.85fr;
      gap: 12px;
      align-items: stretch;
    }

    @media (max-width: 620px) {
      .main-grid {
        grid-template-columns: 1fr;
      }
    }

    .algebra-panel {
      display: flex;
      flex-direction: column;
      gap: 8px;
    }

    .step-card {
      border: 1.5px solid #cbd5e1;
      border-radius: 8px;
      padding: 8px 10px;
      background: #f8fafc;
      transition: all 0.3s ease;
      cursor: pointer;
      color: #1e293b;
    }

    .step-card.active {
      border-color: #0284c7;
      background: #f0f9ff;
      box-shadow: 0 2px 6px rgba(2, 132, 199, 0.18);
    }

    .step-card.inactive {
      background: #ffffff;
      border-color: #e2e8f0;
      color: #334155;
    }

    .step-card:hover {
      border-color: #94a3b8;
    }

    .step-card.active:hover {
      border-color: #0284c7;
    }

    .card-header {
      display: flex;
      align-items: center;
      justify-content: space-between;
      margin-bottom: 4px;
      gap: 6px;
      flex-wrap: wrap;
    }

    .badge {
      font-size: 11px;
      font-weight: 700;
      padding: 2px 7px;
      border-radius: 4px;
      background: #e2e8f0;
      color: #1e293b;
      display: inline-flex;
      align-items: center;
      gap: 4px;
      transition: all 0.3s ease;
    }

    .step-card.active .badge {
      background: #0284c7;
      color: #ffffff;
    }

    .badge-rule {
      font-size: 10px;
      font-weight: 600;
      color: #475569;
      transition: all 0.3s ease;
    }

    .step-card.active .badge-rule {
      color: #0369a1;
    }

    .card-math {
      font-size: 14px;
      line-height: 1.4;
      color: #0f172a;
      display: flex;
      align-items: center;
      flex-wrap: wrap;
      gap: 4px 6px;
      padding: 2px 0;
      font-family: "Cambria Math", "KaTeX_Math", "Times New Roman", serif;
    }

    .math-token {
      font-style: italic;
    }

    .dy-tag {
      font-style: normal;
      font-weight: 700;
      color: #dc2626;
      background: #fee2e2;
      padding: 0 4px;
      border-radius: 3px;
      display: inline-flex;
      align-items: center;
    }

    .frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      font-size: 0.9em;
      line-height: 1.1;
      padding: 0 2px;
    }

    .frac-top {
      border-bottom: 1px solid currentColor;
      padding-bottom: 1px;
    }

    .frac-bot {
      padding-top: 1px;
    }

    .math-res {
      font-weight: 700;
      color: #0284c7;
    }

    .step-card.active .math-res {
      color: #0369a1;
    }

    /* Graph side */
    .graph-panel {
      border: 1.5px solid #cbd5e1;
      border-radius: 8px;
      background: #ffffff;
      padding: 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      position: relative;
      min-height: 240px;
    }

    .graph-svg {
      width: 100%;
      max-width: 250px;
      height: auto;
      aspect-ratio: 1/1;
    }

    .graph-caption {
      margin-top: 6px;
      font-size: 12px;
      font-weight: 700;
      color: #0369a1;
      text-align: center;
      min-height: 18px;
      transition: all 0.3s ease;
    }

    .graph-sub {
      font-size: 11px;
      font-weight: 500;
      color: #475569;
      text-align: center;
      transition: all 0.3s ease;
    }
  </style>
</head>
<body>

  <div class="nav-bar">
    <div class="step-pills">
      <button class="pill active" onclick="setStep(0)">
        <span class="num">1</span> d/dx
      </button>
      <button class="pill" onclick="setStep(1)">
        <span class="num">2</span> Isolate
      </button>
      <button class="pill" onclick="setStep(2)">
        <span class="num">3</span> Solve
      </button>
      <button class="pill" onclick="setStep(3)">
        <span class="num">4</span> (3, 4)
      </button>
    </div>
    <button class="ctrl-btn" id="playBtn" onclick="togglePlay()">⏸ Pause</button>
  </div>

  <div class="main-grid">
    <div class="algebra-panel">
      
      <div class="step-card active" id="card0" onclick="setStep(0)">
        <div class="card-header">
          <span class="badge">Step 1: Differentiate</span>
          <span class="badge-rule">Apply d/dx to both sides</span>
        </div>
        <div class="card-math">
          <span><span class="frac"><span class="frac-top">d</span><span class="frac-bot">dx</span></span>[<span class="math-token">x</span>² + <span class="math-token">y</span>²] = <span class="frac"><span class="frac-top">d</span><span class="frac-bot">dx</span></span>[25]</span>
          <span>&rArr;</span>
          <span>2<span class="math-token">x</span> + 2<span class="math-token">y</span><span class="dy-tag"><span class="frac"><span class="frac-top">dy</span><span class="frac-bot">dx</span></span></span> = 0</span>
        </div>
      </div>

      
      <div class="step-card inactive" id="card1" onclick="setStep(1)">
        <div class="card-header">
          <span class="badge">Step 2: Collect dy/dx</span>
          <span class="badge-rule">Subtract 2x from both sides</span>
        </div>
        <div class="card-math">
          <span>2<span class="math-token">y</span><span class="dy-tag"><span class="frac"><span class="frac-top">dy</span><span class="frac-bot">dx</span></span></span> = &minus;2<span class="math-token">x</span></span>
        </div>
      </div>

      
      <div class="step-card inactive" id="card2" onclick="setStep(2)">
        <div class="card-header">
          <span class="badge">Step 3: Solve for dy/dx</span>
          <span class="badge-rule">Divide by 2y</span>
        </div>
        <div class="card-math">
          <span class="dy-tag"><span class="frac"><span class="frac-top">dy</span><span class="frac-bot">dx</span></span></span>
          <span>=</span>
          <span class="frac"><span class="frac-top">&minus;2<span class="math-token">x</span></span><span class="frac-bot">2<span class="math-token">y</span></span></span>
          <span>=</span>
          <span class="math-res">&minus;<span class="frac"><span class="frac-top"><span class="math-token">x</span></span><span class="frac-bot"><span class="math-token">y</span></span></span></span>
        </div>
      </div>

      
      <div class="step-card inactive" id="card3" onclick="setStep(3)">
        <div class="card-header">
          <span class="badge">Step 4: Slope at (3, 4)</span>
          <span class="badge-rule">Substitute x = 3, y = 4</span>
        </div>
        <div class="card-math">
          <span><span class="math-token">m</span> = <span class="dy-tag"><span class="frac"><span class="frac-top">dy</span><span class="frac-bot">dx</span></span></span><sub style="font-size:0.8em; margin-left:2px;">(3, 4)</sub></span>
          <span>=</span>
          <span class="math-res">&minus;<span class="frac"><span class="frac-top">3</span><span class="frac-bot">4</span></span></span>
          <span style="font-size:12px; color:#475569; margin-left:4px;">(tangent slope)</span>
        </div>
      </div>
    </div>

    
    <div class="graph-panel">
      <svg class="graph-svg" viewBox="-7.5 -7.5 15 15">
        
        <line x1="-6.5" y1="0" x2="6.5" y2="0" stroke="#94a3b8" stroke-width="0.25" />
        <line x1="0" y1="-6.5" x2="0" y2="6.5" stroke="#94a3b8" stroke-width="0.25" />
        
        
        <line x1="5" y1="-0.3" x2="5" y2="0.3" stroke="#64748b" stroke-width="0.25" />
        <line x1="-5" y1="-0.3" x2="-5" y2="0.3" stroke="#64748b" stroke-width="0.25" />
        <line x1="-0.3" y1="5" x2="0.3" y2="5" stroke="#64748b" stroke-width="0.25" />
        <line x1="-0.3" y1="-5" x2="0.3" y2="-5" stroke="#64748b" stroke-width="0.25" />
        
        
        <text x="5.2" y="1.1" font-size="0.85" fill="#475569" font-weight="600" text-anchor="start">5</text>
        <text x="-0.6" y="-4.9" font-size="0.85" fill="#475569" font-weight="600" text-anchor="end">5</text>
        <text x="6.2" y="-0.4" font-size="0.8" fill="#64748b" font-weight="700">x</text>
        <text x="0.4" y="-6.0" font-size="0.8" fill="#64748b" font-weight="700">y</text>

        
        <circle cx="0" cy="0" r="5" fill="none" stroke="#0284c7" stroke-width="0.4" />
        
        
        <line id="radiusLine" x1="0" y1="0" x2="3" y2="-4" stroke="#64748b" stroke-width="0.25" stroke-dasharray="0.5 0.3" opacity="0.5" style="transition: all 0.3s ease;" />

        
        <line id="tangentLine" x1="-1" y1="-7" x2="7" y2="-1" stroke="#dc2626" stroke-width="0.45" opacity="0.25" style="transition: all 0.3s ease;" />

        
        <g id="slopeTriangle" opacity="0" style="transition: opacity 0.3s ease;">
          <line x1="3" y1="-4" x2="7" y2="-4" stroke="#475569" stroke-width="0.25" stroke-dasharray="0.3 0.2" />
          <line x1="7" y1="-4" x2="7" y2="-1" stroke="#dc2626" stroke-width="0.25" stroke-dasharray="0.3 0.2" />
          <text x="5" y="-4.3" font-size="0.75" fill="#334155" font-weight="700" text-anchor="middle">&Delta;x = +4</text>
          <text x="7.3" y="-2.3" font-size="0.75" fill="#dc2626" font-weight="700" text-anchor="start">&Delta;y = &minus;3</text>
        </g>

        
        <circle id="targetPoint" cx="3" cy="-4" r="0.45" fill="#0284c7" stroke="#ffffff" stroke-width="0.15" style="transition: all 0.3s ease;" />
        
        <text id="pointLabel" x="3.4" y="-4.3" font-size="0.9" font-weight="700" fill="#0f172a" text-anchor="start">(3, 4)</text>
      </svg>
      <div class="graph-caption" id="graphCap">Circle: x² + y² = 25 (Radius = 5)</div>
      <div class="graph-sub" id="graphSub">Step 1: Differentiate both sides with respect to x</div>
    </div>
  </div>

  <script>
    const stepsData = [
      {
        caption: "Circle: x² + y² = 25 (Radius = 5)",
        sub: "Step 1: Differentiate both sides with respect to x",
        highlightSlope: false
      },
      {
        caption: "Isolating dy/dx terms",
        sub: "Step 2: 2y(dy/dx) = -2x",
        highlightSlope: false
      },
      {
        caption: "General derivative: dy/dx = -x/y",
        sub: "Step 3: Slope formula for any point (x, y) on circle",
        highlightSlope: false
      },
      {
        caption: "At (3, 4): Tangent slope m = -3/4",
        sub: "Step 4: m = -x/y = -3/4 (Perpendicular to radius 4/3)",
        highlightSlope: true
      }
    ];

    let currentStep = 0;
    let timer = null;
    let isPlaying = true;

    function updateView() {
      // Update pills
      document.querySelectorAll('.pill').forEach((p, idx) => {
        p.classList.toggle('active', idx === currentStep);
      });

      // Update cards
      for (let i = 0; i < 4; i++) {
        const card = document.getElementById('card' + i);
        card.classList.toggle('active', i === currentStep);
        card.classList.toggle('inactive', i !== currentStep);
      }

      // Update graph
      const data = stepsData[currentStep];
      const tangent = document.getElementById('tangentLine');
      const slopeTri = document.getElementById('slopeTriangle');
      const point = document.getElementById('targetPoint');
      const radLine = document.getElementById('radiusLine');
      const cap = document.getElementById('graphCap');
      const sub = document.getElementById('graphSub');

      cap.textContent = data.caption;
      sub.textContent = data.sub;

      if (data.highlightSlope) {
        tangent.setAttribute('opacity', '1');
        tangent.setAttribute('stroke-width', '0.5');
        slopeTri.setAttribute('opacity', '1');
        radLine.setAttribute('opacity', '0.9');
        radLine.setAttribute('stroke', '#0284c7');
        point.setAttribute('fill', '#dc2626');
        point.setAttribute('r', '0.55');
      } else {
        tangent.setAttribute('opacity', '0.2');
        tangent.setAttribute('stroke-width', '0.35');
        slopeTri.setAttribute('opacity', '0');
        radLine.setAttribute('opacity', '0.4');
        radLine.setAttribute('stroke', '#64748b');
        point.setAttribute('fill', '#0284c7');
        point.setAttribute('r', '0.45');
      }
    }

    function setStep(idx) {
      currentStep = idx;
      updateView();
      if (isPlaying) {
        resetTimer();
      }
    }

    function nextStep() {
      currentStep = (currentStep + 1) % 4;
      updateView();
    }

    function resetTimer() {
      if (timer) clearInterval(timer);
      timer = setInterval(nextStep, 3800);
    }

    function togglePlay() {
      isPlaying = !isPlaying;
      const btn = document.getElementById('playBtn');
      if (isPlaying) {
        btn.textContent = '⏸ Pause';
        resetTimer();
      } else {
        btn.textContent = '▶ Play';
        if (timer) clearInterval(timer);
      }
    }

    window.addEventListener('DOMContentLoaded', () => {
      updateView();
      resetTimer();
    });
  </script>
</body>
</html>

Implicit Differentiation for First Derivatives

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Product Rule for Implicit Differentiation</title>
  <style>
    * { 
      box-sizing: border-box; 
      margin: 0; 
      padding: 0; 
    }
    
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8fafc;
      color: #0f172a;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      min-height: 100vh;
      max-height: 590px;
      padding: 12px 10px;
      user-select: none;
    }

    /* Math rendering styling */
    .math {
      font-family: "Cambria Math", "STIX Two Math", "Times New Roman", serif;
      font-size: 1.15rem;
      display: inline-flex;
      align-items: center;
      justify-content: center;
      gap: 2px;
      color: #0f172a;
      line-height: 1.2;
    }

    .frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      font-size: 0.9em;
      line-height: 1;
      padding: 0 2px;
      margin: 0 1px;
    }

    .num {
      border-bottom: 1.8px solid currentColor;
      padding-bottom: 2px;
      display: block;
    }

    .den {
      padding-top: 2px;
      display: block;
    }

    .var-u {
      color: #1d4ed8;
      font-weight: 700;
      font-style: italic;
    }

    .var-v {
      color: #7e22ce;
      font-weight: 700;
      font-style: italic;
    }

    .op {
      color: #334155;
      font-weight: 600;
      margin: 0 2px;
    }

    .bracket {
      font-size: 1.25em;
      color: #334155;
      font-weight: 300;
    }

    /* Legend Header */
    .legend {
      display: flex;
      gap: 12px;
      margin-bottom: 6px;
      font-size: 13px;
      font-weight: 600;
    }
    
    .legend-item {
      display: flex;
      align-items: center;
      gap: 6px;
      padding: 4px 10px;
      border-radius: 6px;
      background: #ffffff;
      border: 1.5px solid #cbd5e1;
      color: #1e293b;
    }
    
    .legend-dot {
      width: 10px;
      height: 10px;
      border-radius: 50%;
    }
    .dot-u { background: #1d4ed8; }
    .dot-v { background: #7e22ce; }

    /* Diagram Tree */
    .tree-container {
      position: relative;
      width: 100%;
      max-width: 480px;
      display: flex;
      flex-direction: column;
      align-items: center;
      flex-grow: 1;
      justify-content: center;
      gap: 8px;
    }

    /* Node Cards */
    .node {
      background: #ffffff;
      border: 2px solid #cbd5e1;
      border-radius: 8px;
      padding: 6px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      transition: border-color 0.25s ease, box-shadow 0.25s ease, opacity 0.25s ease;
      z-index: 2;
    }

    .node-title {
      font-size: 11px;
      text-transform: uppercase;
      letter-spacing: 0.6px;
      color: #334155;
      margin-bottom: 3px;
      font-weight: 700;
    }

    .root-node {
      width: 100%;
      max-width: 280px;
      border-color: #94a3b8;
      background: #ffffff;
    }

    .root-node.active-focus {
      border-color: #0284c7;
      box-shadow: 0 0 0 3px rgba(2, 132, 199, 0.2);
    }

    /* Branch row */
    .branches-row {
      display: flex;
      justify-content: space-between;
      width: 100%;
      gap: 8px;
      position: relative;
    }

    .branch-col {
      flex: 1;
      display: flex;
      flex-direction: column;
      gap: 6px;
    }

    .sub-node {
      padding: 6px 8px;
      min-height: 52px;
    }

    .branch-left {
      border-left: 4px solid #1d4ed8;
      background: #f8faff;
    }
    .branch-left.active-focus {
      border-color: #1d4ed8;
      box-shadow: 0 0 0 3px rgba(29, 78, 216, 0.2);
      background: #ffffff;
    }

    .branch-right {
      border-right: 4px solid #7e22ce;
      background: #faf7fd;
    }
    .branch-right.active-focus {
      border-color: #7e22ce;
      box-shadow: 0 0 0 3px rgba(126, 34, 206, 0.2);
      background: #ffffff;
    }

    .result-node {
      background: #f0fdf4;
      border: 2px solid #16a34a;
      width: 100%;
      max-width: 320px;
      padding: 8px 12px;
    }

    .result-node.active-focus {
      box-shadow: 0 0 0 3px rgba(22, 163, 74, 0.25);
    }

    /* Badge */
    .badge-chain {
      font-size: 10px;
      font-weight: 700;
      color: #854d0e;
      background: #fef9c3;
      border: 1px solid #ca8a04;
      padding: 1px 6px;
      border-radius: 10px;
      margin-top: 3px;
      display: inline-block;
    }

    /* Step Navigation Controls */
    .controls {
      display: flex;
      align-items: center;
      gap: 12px;
      margin-top: 4px;
    }
    
    .btn {
      background: #ffffff;
      border: 1.5px solid #64748b;
      color: #0f172a;
      padding: 5px 12px;
      border-radius: 6px;
      font-size: 13px;
      font-weight: 700;
      cursor: pointer;
      display: flex;
      align-items: center;
      gap: 4px;
      transition: background 0.15s, border-color 0.15s;
    }
    .btn:active { 
      background: #e2e8f0; 
    }
    
    .btn-primary {
      background: #0f172a;
      color: #ffffff;
      border-color: #0f172a;
    }
    .btn-primary:active { 
      background: #334155; 
    }

    .dots-container {
      display: flex;
      gap: 8px;
    }
    
    .dot {
      width: 10px;
      height: 10px;
      border-radius: 50%;
      background: #94a3b8;
      cursor: pointer;
      transition: background 0.2s;
    }
    .dot.active {
      background: #0f172a;
    }

    /* Visibility transitions */
    .fade-step {
      opacity: 0.35;
      transition: opacity 0.25s ease;
    }
    .fade-step.visible {
      opacity: 1;
    }

    /* Small screens adjustments */
    @media (max-width: 380px) {
      .math { font-size: 1rem; }
      .legend { font-size: 11px; gap: 6px; }
      .node-title { font-size: 10px; }
      .sub-node { padding: 4px 6px; }
      .badge-chain { font-size: 9px; }
    }
  </style>
</head>
<body>

  
  <div class="legend">
    <div class="legend-item">
      <div class="legend-dot dot-u"></div>
      <span>Let <span class="var-u">u</span> = <span class="var-u">x</span></span>
    </div>
    <div class="legend-item">
      <div class="legend-dot dot-v"></div>
      <span>Let <span class="var-v">v</span> = <span class="var-v">y</span></span>
    </div>
  </div>

  
  <div class="tree-container" id="tree">
    
    
    <div class="node root-node" id="nodeRoot">
      <div class="node-title">Product Rule: <span style="font-weight:800; color:#0f172a;">d/dx [ u · v ]</span></div>
      <div class="math">
        <span class="frac"><span class="num">d</span><span class="den">dx</span></span>
        <span class="bracket">[</span>
        <span class="var-u">x</span>
        <span class="op">·</span>
        <span class="var-v">y</span>
        <span class="bracket">]</span>
      </div>
    </div>

    
    <div class="branches-row">
      
      <div class="branch-col">
        <div class="node sub-node branch-left fade-step" id="nodeLeft1">
          <div class="node-title" style="color:#1d4ed8;">Branch 1: u' · v</div>
          <div class="math">
            <span class="frac"><span class="num">d</span><span class="den">dx</span></span>
            <span class="bracket">[</span><span class="var-u">x</span><span class="bracket">]</span>
            <span class="op">·</span>
            <span class="var-v">y</span>
          </div>
        </div>
        
        <div class="node sub-node branch-left fade-step" id="nodeLeft2">
          <div class="node-title" style="color:#1d4ed8;">Since d/dx[x] = 1</div>
          <div class="math">
            <span>(1)</span>
            <span class="op">·</span>
            <span class="var-v">y</span>
            <span class="op">=</span>
            <span class="var-v">y</span>
          </div>
        </div>
      </div>

      
      <div class="branch-col">
        <div class="node sub-node branch-right fade-step" id="nodeRight1">
          <div class="node-title" style="color:#7e22ce;">Branch 2: u · v'</div>
          <div class="math">
            <span class="var-u">x</span>
            <span class="op">·</span>
            <span class="frac"><span class="num">d</span><span class="den">dx</span></span>
            <span class="bracket">[</span><span class="var-v">y</span><span class="bracket">]</span>
          </div>
        </div>
        
        <div class="node sub-node branch-right fade-step" id="nodeRight2">
          <div class="node-title" style="color:#7e22ce;">Chain Rule on y</div>
          <div class="math">
            <span class="var-u">x</span>
            <span class="op">·</span>
            <span class="frac" style="font-weight:700;"><span class="num">d<span class="var-v">y</span></span><span class="den">dx</span></span>
          </div>
          <div class="badge-chain">Implicit: dy/dx</div>
        </div>
      </div>
    </div>

    
    <div class="node result-node fade-step" id="nodeResult">
      <div class="node-title" style="color: #15803d;">Sum Branches: u'v + uv'</div>
      <div class="math" style="font-size: 1.25rem;">
        <span class="var-v">y</span>
        <span class="op" style="font-size:1.1em; font-weight:700;">+</span>
        <span class="var-u">x</span>
        <span class="frac" style="font-weight:700; margin-left: 2px;"><span class="num">d<span class="var-v">y</span></span><span class="den">dx</span></span>
      </div>
    </div>

  </div>

  
  <div class="controls">
    <button class="btn" id="prevBtn" title="Previous step">← Back</button>
    <div class="dots-container" id="dotsContainer">
      <div class="dot active" title="Step 1"></div>
      <div class="dot" title="Step 2"></div>
      <div class="dot" title="Step 3"></div>
      <div class="dot" title="Step 4"></div>
    </div>
    <button class="btn btn-primary" id="nextBtn" title="Next step">Next →</button>
    <button class="btn" id="playBtn" style="padding: 5px 8px;" title="Auto play / pause">⏸</button>
  </div>

  <script>
    let currentStep = 0;
    const totalSteps = 4;
    let autoPlayInterval = null;
    let isPlaying = true;

    const elLeft1 = document.getElementById('nodeLeft1');
    const elLeft2 = document.getElementById('nodeLeft2');
    const elRight1 = document.getElementById('nodeRight1');
    const elRight2 = document.getElementById('nodeRight2');
    const elResult = document.getElementById('nodeResult');
    const elRoot = document.getElementById('nodeRoot');
    const dots = document.querySelectorAll('.dot');
    const playBtn = document.getElementById('playBtn');

    const updateStep = (step) => {
      currentStep = step;

      // Update dots
      dots.forEach((d, i) => {
        d.classList.toggle('active', i === currentStep);
      });

      // Clear focus & visibility
      [elLeft1, elLeft2, elRight1, elRight2, elResult].forEach(el => {
        el.classList.remove('visible', 'active-focus');
      });
      elRoot.classList.remove('active-focus');

      // Progressive reveal based on step
      if (currentStep === 0) {
        elRoot.classList.add('active-focus');
      } else if (currentStep === 1) {
        elLeft1.classList.add('visible', 'active-focus');
        elRight1.classList.add('visible', 'active-focus');
      } else if (currentStep === 2) {
        elLeft1.classList.add('visible');
        elRight1.classList.add('visible');
        elLeft2.classList.add('visible', 'active-focus');
        elRight2.classList.add('visible', 'active-focus');
      } else if (currentStep === 3) {
        elLeft1.classList.add('visible');
        elRight1.classList.add('visible');
        elLeft2.classList.add('visible');
        elRight2.classList.add('visible');
        elResult.classList.add('visible', 'active-focus');
      }
    };

    const nextStep = () => {
      updateStep((currentStep + 1) % totalSteps);
    };

    const prevStep = () => {
      updateStep((currentStep - 1 + totalSteps) % totalSteps);
    };

    const startAutoPlay = () => {
      if (autoPlayInterval) clearInterval(autoPlayInterval);
      autoPlayInterval = setInterval(nextStep, 3200);
      isPlaying = true;
      playBtn.textContent = '⏸';
    };

    const stopAutoPlay = () => {
      if (autoPlayInterval) clearInterval(autoPlayInterval);
      autoPlayInterval = null;
      isPlaying = false;
      playBtn.textContent = '▶';
    };

    // Event Listeners
    document.getElementById('nextBtn').addEventListener('click', () => {
      stopAutoPlay();
      nextStep();
    });

    document.getElementById('prevBtn').addEventListener('click', () => {
      stopAutoPlay();
      prevStep();
    });

    dots.forEach((dot, index) => {
      dot.addEventListener('click', () => {
        stopAutoPlay();
        updateStep(index);
      });
    });

    playBtn.addEventListener('click', () => {
      if (isPlaying) {
        stopAutoPlay();
      } else {
        nextStep();
        startAutoPlay();
      }
    });

    // Start
    updateStep(0);
    startAutoPlay();
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Implicit Differentiation - Folium of Descartes</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #ffffff;
      color: #0f172a;
      padding: 10px;
      min-height: 100%;
    }
    .layout {
      display: grid;
      grid-template-columns: 1.15fr 1fr;
      gap: 14px;
      align-items: center;
      max-width: 760px;
      margin: 0 auto;
    }
    @media (max-width: 580px) {
      .layout {
        grid-template-columns: 1fr;
        gap: 12px;
      }
    }
    .section {
      display: flex;
      flex-direction: column;
      gap: 8px;
    }
    .header-bar {
      display: flex;
      justify-content: space-between;
      align-items: center;
    }
    .badge {
      display: inline-flex;
      align-items: center;
      padding: 3px 10px;
      background: #f1f5f9;
      border: 1px solid #e2e8f0;
      border-radius: 20px;
      font-size: 11px;
      font-weight: 600;
      color: #475569;
    }
    .badge.accent {
      background: #eff6ff;
      border-color: #bfdbfe;
      color: #1d4ed8;
      font-weight: 700;
    }
    .step-nav {
      display: flex;
      gap: 6px;
      justify-content: space-between;
    }
    .step-btn {
      flex: 1;
      padding: 6px 2px;
      font-size: 11px;
      font-weight: 600;
      border: 1px solid #cbd5e1;
      background: #f8fafc;
      border-radius: 6px;
      cursor: pointer;
      color: #475569;
      text-align: center;
      transition: background 0.15s ease, color 0.15s ease, border-color 0.15s ease;
    }
    .step-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
    }
    .math-display-area {
      min-height: 85px;
      display: flex;
      flex-direction: column;
      justify-content: center;
      align-items: center;
      background: #f8fafc;
      border-radius: 8px;
      padding: 10px 8px;
      border: 1px dashed #cbd5e1;
    }
    .step-title {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
      margin-bottom: 6px;
      text-align: center;
    }
    .math-row {
      font-family: "Cambria Math", "Latin Modern Math", "STIX Two Math", "Times New Roman", Georgia, serif;
      font-size: 15px;
      color: #0f172a;
      display: flex;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
      gap: 3px;
      line-height: 1.3;
    }
    sup {
      font-size: 0.72em;
      line-height: 0;
      vertical-align: 0.45em;
    }
    sub {
      font-size: 0.72em;
      line-height: 0;
      vertical-align: -0.3em;
    }
    .frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      padding: 0 2px;
      font-size: 0.9em;
    }
    .frac .num {
      border-bottom: 1.2px solid #0f172a;
      padding-bottom: 1px;
    }
    .frac .den {
      padding-top: 1px;
    }
    .var {
      font-style: italic;
    }
    .num-val {
      font-style: normal;
    }
    .highlight-box {
      background: #eff6ff;
      border-left: 3px solid #2563eb;
      padding: 6px 10px;
      border-radius: 4px;
      font-size: 12px;
      color: #1e3a8a;
      line-height: 1.35;
    }
    .controls {
      display: flex;
      gap: 8px;
      justify-content: center;
    }
    .nav-arrow {
      padding: 4px 14px;
      font-size: 12px;
      border: 1px solid #cbd5e1;
      background: #ffffff;
      border-radius: 6px;
      cursor: pointer;
      font-weight: 600;
      color: #334155;
      transition: background 0.15s ease;
    }
    .nav-arrow:hover {
      background: #f1f5f9;
    }
    .graph-wrapper {
      position: relative;
      width: 100%;
      aspect-ratio: 1;
      max-height: 250px;
      display: flex;
      align-items: center;
      justify-content: center;
    }
    svg {
      width: 100%;
      height: 100%;
      overflow: hidden;
    }
  </style>
</head>
<body>

  <div class="layout">
    
    <div class="section">
      <div class="header-bar">
        <span class="badge accent"><span class="var">x</span><sup>3</sup> + <span class="var">y</span><sup>3</sup> = 6<span class="var">xy</span></span>
        <span id="stepCounter" class="badge">Step 1 of 4</span>
      </div>

      <div class="step-nav">
        <button class="step-btn active" onclick="setStep(0)">1. Diff</button>
        <button class="step-btn" onclick="setStep(1)">2. Group</button>
        <button class="step-btn" onclick="setStep(2)">3. Divide</button>
        <button class="step-btn" onclick="setStep(3)">4. Slope</button>
      </div>

      <div class="math-display-area">
        <div id="stepLabel" class="step-title">1. Differentiate both sides</div>
        <div id="mathDisplay" class="math-row"></div>
      </div>

      <div id="stepExplanation" class="highlight-box">
        Apply power rule &amp; product rule: <span class="frac"><span class="num">d</span><span class="den">d<span class="var">x</span></span></span>(6<span class="var">xy</span>) = 6<span class="var">y</span> + 6<span class="var">x</span>(<span class="frac"><span class="num">d<span class="var">y</span></span><span class="den">d<span class="var">x</span></span></span>)
      </div>

      <div class="controls">
        <button class="nav-arrow" onclick="prevStep()">← Prev</button>
        <button class="nav-arrow" onclick="nextStep()">Next →</button>
      </div>
    </div>

    
    <div class="section" style="align-items: center;">
      <div class="header-bar" style="width: 100%;">
        <span style="font-size: 11px; font-weight: 700; color: #64748b; letter-spacing: 0.5px;">FOLIUM OF DESCARTES</span>
        <span id="pointCoord" style="font-size: 12px; font-weight: 700; color: #2563eb;">Point (3, 3)</span>
      </div>
      <div class="graph-wrapper">
        <svg id="plotSvg" viewBox="0 0 200 200">
          <defs>
            <clipPath id="plotClip">
              <rect x="5" y="5" width="190" height="190" rx="4" />
            </clipPath>
          </defs>

          
          <g stroke="#f1f5f9" stroke-width="1">
            <line x1="20" y1="10" x2="20" y2="190" />
            <line x1="60" y1="10" x2="60" y2="190" />
            <line x1="80" y1="10" x2="80" y2="190" />
            <line x1="100" y1="10" x2="100" y2="190" />
            <line x1="120" y1="10" x2="120" y2="190" />
            <line x1="140" y1="10" x2="140" y2="190" />
            <line x1="160" y1="10" x2="160" y2="190" />
            <line x1="10" y1="40" x2="190" y2="40" />
            <line x1="10" y1="60" x2="190" y2="60" />
            <line x1="10" y1="80" x2="190" y2="80" />
            <line x1="10" y1="100" x2="190" y2="100" />
            <line x1="10" y1="120" x2="190" y2="120" />
            <line x1="10" y1="140" x2="190" y2="140" />
            <line x1="10" y1="180" x2="190" y2="180" />
          </g>

          
          <line x1="10" y1="160" x2="188" y2="160" stroke="#94a3b8" stroke-width="1.2" />
          <line x1="40" y1="190" x2="40" y2="12" stroke="#94a3b8" stroke-width="1.2" />

          
          <polygon points="190,160 184,157.5 184,162.5" fill="#94a3b8" />
          <polygon points="40,10 37.5,16 42.5,16" fill="#94a3b8" />
          <text x="186" y="154" font-size="9" font-style="italic" fill="#64748b" font-weight="bold">x</text>
          <text x="46" y="16" font-size="9" font-style="italic" fill="#64748b" font-weight="bold">y</text>

          
          <line x1="100" y1="157" x2="100" y2="163" stroke="#64748b" stroke-width="1.2" />
          <text x="100" y="172" font-size="8.5" fill="#64748b" text-anchor="middle" font-weight="600">3</text>

          <line x1="160" y1="157" x2="160" y2="163" stroke="#94a3b8" stroke-width="1" />
          <text x="160" y="172" font-size="8" fill="#94a3b8" text-anchor="middle">6</text>

          
          <line x1="37" y1="100" x2="43" y2="100" stroke="#64748b" stroke-width="1.2" />
          <text x="33" y="103" font-size="8.5" fill="#64748b" text-anchor="end" font-weight="600">3</text>

          <line x1="37" y1="40" x2="43" y2="40" stroke="#94a3b8" stroke-width="1" />
          <text x="33" y="43" font-size="8" fill="#94a3b8" text-anchor="end">6</text>

          
          <text x="34" y="171" font-size="8" fill="#94a3b8" text-anchor="end">0</text>

          
          <line x1="100" y1="160" x2="100" y2="100" stroke="#93c5fd" stroke-width="1" stroke-dasharray="3 2" />
          <line x1="40" y1="100" x2="100" y2="100" stroke="#93c5fd" stroke-width="1" stroke-dasharray="3 2" />

          
          <g clip-path="url(#plotClip)">
            
            <path id="curveBranchQ2" fill="none" stroke="#2563eb" stroke-width="1.8" opacity="0.6" stroke-linecap="round" />
            <path id="curveBranchQ4" fill="none" stroke="#2563eb" stroke-width="1.8" opacity="0.6" stroke-linecap="round" />
            
            <path id="curveLoop" fill="none" stroke="#2563eb" stroke-width="2.2" stroke-linejoin="round" stroke-linecap="round" />

            
            <line id="tangentLine" x1="45" y1="45" x2="155" y2="155" stroke="#ef4444" stroke-width="1.8" stroke-dasharray="4 3" opacity="0.25" style="transition: opacity 0.3s ease;" />
          </g>

          
          <g id="tangentTag" transform="translate(108, 68)" opacity="0" style="transition: opacity 0.3s ease;">
            <rect x="0" y="-12" width="46" height="17" rx="4" fill="#fee2e2" stroke="#ef4444" stroke-width="0.8"/>
            <text x="23" y="0" font-size="9" font-weight="bold" fill="#b91c1c" text-anchor="middle">m = −1</text>
          </g>

          
          <circle id="pointDot" cx="100" cy="100" r="4" fill="#ef4444" stroke="#ffffff" stroke-width="1.5" />
          <text x="100" y="90" font-size="9" font-weight="bold" fill="#1e293b" text-anchor="middle">(3, 3)</text>
        </svg>
      </div>
    </div>
  </div>

  <script>
    // Coordinate mapping: origin (0,0) at SVG (40, 160), scale 20 px/unit
    function toSvg(x, y) {
      return [40 + x * 20, 160 - y * 20];
    }

    // Exact Parametric Folium of Descartes: x(t) = 6t / (1 + t^3), y(t) = 6t^2 / (1 + t^3)
    function buildFolium() {
      // 1. Loop: t in [0, 50], smooth continuous loop passing through (3, 3) at t = 1
      const loopPts = [];
      const loopSteps = 240;
      for (let i = 0; i <= loopSteps; i++) {
        const u = i / loopSteps;
        // Non-linear distribution to sample densely around t = 1 and close back at origin smoothly
        const t = u === 1 ? 80 : Math.pow(u / (1 - u + 0.001), 1.5);
        const denom = 1 + t * t * t;
        const x = (6 * t) / denom;
        const y = (6 * t * t) / denom;
        const [sx, sy] = toSvg(x, y);
        loopPts.push(`${i === 0 ? 'M' : 'L'} ${sx.toFixed(2)} ${sy.toFixed(2)}`);
      }
      // Explicitly close loop at origin
      const [ox, oy] = toSvg(0, 0);
      loopPts.push(`L ${ox.toFixed(2)} ${oy.toFixed(2)} Z`);
      document.getElementById('curveLoop').setAttribute('d', loopPts.join(' '));

      // 2. Quadrant II Branch: x < 0, y > 0 for t in (-1, 0]
      const q2Pts = [];
      const q2Steps = 100;
      for (let i = 0; i <= q2Steps; i++) {
        // Sample from 0 down to -0.86 (approaches asymptote x + y + 2 = 0)
        const t = 0 - (0.86 * (i / q2Steps));
        const denom = 1 + t * t * t;
        const x = (6 * t) / denom;
        const y = (6 * t * t) / denom;
        const [sx, sy] = toSvg(x, y);
        q2Pts.push(`${i === 0 ? 'M' : 'L'} ${sx.toFixed(2)} ${sy.toFixed(2)}`);
      }
      document.getElementById('curveBranchQ2').setAttribute('d', q2Pts.join(' '));

      // 3. Quadrant IV Branch: x > 0, y < 0 for t in (-infinity, -1)
      const q4Pts = [];
      const q4Steps = 100;
      for (let i = 0; i <= q4Steps; i++) {
        // Sample from -40 up to -1.16
        const u = i / q4Steps;
        const t = -40 + (40 - 1.16) * Math.pow(u, 2);
        const denom = 1 + t * t * t;
        const x = (6 * t) / denom;
        const y = (6 * t * t) / denom;
        const [sx, sy] = toSvg(x, y);
        q4Pts.push(`${i === 0 ? 'M' : 'L'} ${sx.toFixed(2)} ${sy.toFixed(2)}`);
      }
      document.getElementById('curveBranchQ4').setAttribute('d', q4Pts.join(' '));
    }

    buildFolium();

    const stepData = [
      {
        title: "1. Differentiate both sides w.r.t. x",
        mathHtml: `
          <span class="num-val">3</span><span class="var">x</span><sup>2</sup>
          <span>+</span>
          <span class="num-val">3</span><span class="var">y</span><sup>2</sup><span class="frac"><span class="num">d<span class="var">y</span></span><span class="den">d<span class="var">x</span></span></span>
          <span>=</span>
          <span class="num-val">6</span><span class="var">y</span>
          <span>+</span>
          <span class="num-val">6</span><span class="var">x</span><span class="frac"><span class="num">d<span class="var">y</span></span><span class="den">d<span class="var">x</span></span></span>
        `,
        expl: "Apply power rule to <span class='var'>x</span><sup>3</sup> &amp; <span class='var'>y</span><sup>3</sup>, and product rule to 6<span class='var'>xy</span>: (6<span class='var'>x</span>)'<span class='var'>y</span> + 6<span class='var'>x</span>(<span class='var'>y</span>').",
        showTangent: false
      },
      {
        title: "2. Collect dy/dx terms on one side",
        mathHtml: `
          <span>(</span><span class="num-val">3</span><span class="var">y</span><sup>2</sup>
          <span>−</span>
          <span class="num-val">6</span><span class="var">x</span><span>)</span><span class="frac"><span class="num">d<span class="var">y</span></span><span class="den">d<span class="var">x</span></span></span>
          <span>=</span>
          <span class="num-val">6</span><span class="var">y</span>
          <span>−</span>
          <span class="num-val">3</span><span class="var">x</span><sup>2</sup>
        `,
        expl: "Group all terms containing <span class='frac'><span class='num'>dy</span><span class='den'>dx</span></span> on the left side and remaining terms on the right.",
        showTangent: false
      },
      {
        title: "3. Factor & Divide to isolate dy/dx",
        mathHtml: `
          <span class="frac"><span class="num">d<span class="var">y</span></span><span class="den">d<span class="var">x</span></span></span>
          <span>=</span>
          <span class="frac"><span class="num"><span class="num-val">6</span><span class="var">y</span> − <span class="num-val">3</span><span class="var">x</span><sup>2</sup></span><span class="den"><span class="num-val">3</span><span class="var">y</span><sup>2</sup> − <span class="num-val">6</span><span class="var">x</span></span></span>
          <span>=</span>
          <span class="frac"><span class="num"><span class="num-val">2</span><span class="var">y</span> − <span class="var">x</span><sup>2</sup></span><span class="den"><span class="var">y</span><sup>2</sup> − <span class="num-val">2</span><span class="var">x</span></span></span>
        `,
        expl: "Divide both sides by (3<span class='var'>y</span><sup>2</sup> − 6<span class='var'>x</span>), then divide numerator and denominator by 3 to simplify.",
        showTangent: false
      },
      {
        title: "4. Evaluate slope at (3, 3)",
        mathHtml: `
          <span class="frac"><span class="num">d<span class="var">y</span></span><span class="den">d<span class="var">x</span></span></span><span style="font-size:1.1em; font-style:normal; margin:0 2px;">|</span><sub style="font-size:0.75em; vertical-align:sub; font-style:normal;">(3,3)</sub>
          <span>=</span>
          <span class="frac"><span class="num"><span class="num-val">2(3)</span> − <span class="num-val">3</span><sup>2</sup></span><span class="den"><span class="num-val">3</span><sup>2</sup> − <span class="num-val">2(3)</span></span></span>
          <span>=</span>
          <span class="frac"><span class="num"><span class="num-val">6 − 9</span></span><span class="den"><span class="num-val">9 − 6</span></span></span>
          <span>=</span>
          <span class="num-val" style="font-weight:700; color:#b91c1c;">−1</span>
        `,
        expl: "Substitute <span class='var'>x</span> = 3 and <span class='var'>y</span> = 3 into <span class='frac'><span class='num'>dy</span><span class='den'>dx</span></span>: tangent slope <span class='var'>m</span> = −1.",
        showTangent: true
      }
    ];

    let currentStep = 0;

    function renderStep(idx) {
      currentStep = idx;
      const s = stepData[idx];
      document.getElementById('stepCounter').textContent = `Step ${idx + 1} of ${stepData.length}`;
      document.getElementById('stepLabel').textContent = s.title;
      document.getElementById('mathDisplay').innerHTML = s.mathHtml;
      document.getElementById('stepExplanation').innerHTML = s.expl;

      const btns = document.querySelectorAll('.step-btn');
      btns.forEach((btn, i) => {
        btn.classList.toggle('active', i === idx);
      });

      const tanLine = document.getElementById('tangentLine');
      const tanTag = document.getElementById('tangentTag');
      if (s.showTangent) {
        tanLine.style.opacity = "1";
        tanTag.style.opacity = "1";
      } else {
        tanLine.style.opacity = "0.2";
        tanTag.style.opacity = "0";
      }
    }

    function setStep(i) { renderStep(i); }
    function nextStep() { renderStep((currentStep + 1) % stepData.length); }
    function prevStep() { renderStep((currentStep - 1 + stepData.length) % stepData.length); }

    renderStep(0);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, sans-serif;
      background: #f8fafc;
      color: #1e293b;
      padding: 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100vh;
    }

    .comparison-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 10px;
      width: 100%;
      max-width: 640px;
    }

    @media (max-width: 500px) {
      .comparison-grid {
        grid-template-columns: 1fr;
      }
    }

    .panel {
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 10px;
      padding: 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      box-shadow: 0 1px 3px rgba(0,0,0,0.05);
    }

    .header-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      margin-bottom: 4px;
    }

    .badge {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      padding: 2px 7px;
      border-radius: 6px;
    }

    .badge-blue { background: #e0f2fe; color: #0284c7; }
    .badge-purple { background: #f3e8ff; color: #7e22ce; }

    .formula {
      font-size: 14px;
      font-weight: 700;
      font-family: "SF Pro Text", -apple-system, monospace;
    }

    .sub-tag {
      font-size: 11px;
      color: #64748b;
      margin-bottom: 6px;
      align-self: flex-start;
      display: flex;
      align-items: center;
      gap: 4px;
    }

    .graph-box {
      width: 100%;
      aspect-ratio: 1.15;
      max-height: 180px;
      position: relative;
      touch-action: none;
      user-select: none;
    }

    svg {
      width: 100%;
      height: 100%;
      display: block;
    }

    .stat-pill {
      margin-top: 6px;
      font-size: 11px;
      font-weight: 600;
      padding: 3px 8px;
      border-radius: 16px;
      background: #f1f5f9;
      color: #334155;
      text-align: center;
      width: 100%;
      white-space: nowrap;
      overflow: hidden;
      text-overflow: ellipsis;
    }

    .highlight-purple { color: #9333ea; font-weight: 700; }
    .highlight-blue { color: #0284c7; font-weight: 700; }

    .interactive-hint {
      font-size: 10px;
      color: #94a3b8;
      margin-top: 3px;
      text-align: center;
    }
  </style>
</head>
<body>

  <div class="comparison-grid">
    
    <div class="panel">
      <div class="header-row">
        <span class="badge badge-blue">Explicit</span>
        <span class="formula">y = x² − 4</span>
      </div>
      <div class="sub-tag"><span>✓</span> y alone on one side</div>

      <div class="graph-box">
        <svg viewBox="-5.5 -5.8 11 11" id="svg-explicit">
          
          <line x1="-5" y1="0" x2="5" y2="0" stroke="#cbd5e1" stroke-width="0.15" stroke-dasharray="0.3,0.3" />
          <line x1="0" y1="-5.2" x2="0" y2="4.8" stroke="#cbd5e1" stroke-width="0.15" stroke-dasharray="0.3,0.3" />
          <line x1="-5" y1="0" x2="5" y2="0" stroke="#94a3b8" stroke-width="0.2" />
          <line x1="0" y1="-5.2" x2="0" y2="4.8" stroke="#94a3b8" stroke-width="0.2" />

          
          <path id="parabola-path" fill="none" stroke="#0284c7" stroke-width="0.35" stroke-linecap="round" />

          
          <line x1="2.2" y1="-5" x2="2.2" y2="4.5" stroke="#f59e0b" stroke-width="0.18" stroke-dasharray="0.35,0.3" />
          
          
          <circle cx="2.2" cy="-0.84" r="0.4" fill="#f59e0b" />
          <circle cx="2.2" cy="-0.84" r="0.18" fill="#ffffff" />
          
          
          <text x="1.8" y="-1.3" text-anchor="end" font-size="0.68" fill="#d97706" font-weight="700">1 input → 1 output</text>
          <text x="2.6" y="-0.7" font-size="0.62" fill="#64748b">(2.2, 0.8)</text>
        </svg>
      </div>

      <div class="stat-pill">
        Function: <span class="highlight-blue">y = f(x)</span>
      </div>
      <div class="interactive-hint">Passes vertical line test</div>
    </div>

    
    <div class="panel">
      <div class="header-row">
        <span class="badge badge-purple">Implicit</span>
        <span class="formula">x² + y² = 25</span>
      </div>
      <div class="sub-tag"><span>✦</span> x, y intertwined</div>

      <div class="graph-box" id="circle-container" style="cursor: crosshair;">
        <svg viewBox="-7.5 -7 15 14" id="svg-implicit">
          
          <line x1="-7" y1="0" x2="7" y2="0" stroke="#94a3b8" stroke-width="0.25" />
          <line x1="0" y1="-6.5" x2="0" y2="6.5" stroke="#94a3b8" stroke-width="0.25" />

          
          <circle cx="0" cy="0" r="5" fill="none" stroke="#a855f7" stroke-width="0.4" />

          
          <line id="test-line" x1="3" y1="-6.5" x2="3" y2="6.5" stroke="#f59e0b" stroke-width="0.18" stroke-dasharray="0.35,0.35" />
          
          
          <circle id="point-bottom" cx="3" cy="4" r="0.4" fill="#94a3b8" />
          <text id="label-bottom" x="3.4" y="4.3" font-size="0.8" fill="#64748b">(3, −4)</text>

          
          <line id="tangent-line" stroke="#e11d48" stroke-width="0.32" stroke-linecap="round" />

          
          <circle id="point-p" cx="3" cy="-4" r="0.5" fill="#9333ea" />
          <circle id="point-p-core" cx="3" cy="-4" r="0.2" fill="#ffffff" />
          
          <text id="label-top" x="3.4" y="-4.3" font-size="0.85" fill="#9333ea" font-weight="700">(3, 4)</text>
        </svg>
      </div>

      <div class="stat-pill" id="slope-pill">
        Slope at (3, 4): <span class="highlight-purple" id="slope-text">m = −3/4</span>
      </div>
      <div class="interactive-hint">Drag or tap circle to move point</div>
    </div>
  </div>

  <script>
    // Draw explicit parabola y = x^2 - 4
    const parabola = document.getElementById('parabola-path');
    let d = '';
    for (let x = -2.9; x <= 2.9; x += 0.1) {
      const y = (x * x) - 4;
      const svgY = -y;
      d += (d === '' ? 'M ' : ' L ') + x.toFixed(2) + ' ' + svgY.toFixed(2);
    }
    parabola.setAttribute('d', d);

    // Implicit interactive circle logic
    let angle = Math.atan2(4, 3); // Initial point (3, 4)
    const R = 5;

    function updateTangent() {
      const x = R * Math.cos(angle);
      const y = R * Math.sin(angle);
      const svgY = -y; // Invert for SVG coordinates

      // Update primary interactive point
      const pt = document.getElementById('point-p');
      const ptCore = document.getElementById('point-p-core');
      pt.setAttribute('cx', x);
      pt.setAttribute('cy', svgY);
      ptCore.setAttribute('cx', x);
      ptCore.setAttribute('cy', svgY);

      // Label positioning for active point
      const lbl = document.getElementById('label-top');
      const xOffset = x >= 0 ? 0.45 : -3.2;
      const yOffset = svgY < 0 ? -0.4 : 0.9;
      lbl.setAttribute('x', x + xOffset);
      lbl.setAttribute('y', svgY + yOffset);
      lbl.textContent = `(${x.toFixed(1)}, ${y.toFixed(1)})`;

      // Update vertical line (2 outputs for 1 input test)
      const tLine = document.getElementById('test-line');
      tLine.setAttribute('x1', x);
      tLine.setAttribute('x2', x);

      // Update opposite vertical point (demonstrating two y-values for one x-value)
      const ptBottom = document.getElementById('point-bottom');
      const lblBottom = document.getElementById('label-bottom');
      ptBottom.setAttribute('cx', x);
      ptBottom.setAttribute('cy', -svgY);
      lblBottom.setAttribute('x', x + xOffset);
      lblBottom.setAttribute('y', -svgY + (svgY < 0 ? 0.9 : -0.4));
      lblBottom.textContent = `(${x.toFixed(1)}, ${(-y).toFixed(1)})`;

      // Correct tangent line math:
      // In standard Cartesian coordinates:
      // Position vector: r = (x, y) = (R*cos(θ), R*sin(θ))
      // Tangent vector: T = (-sin(θ), cos(θ)), Cartesian slope m = -x/y = -cos(θ)/sin(θ)
      // In SVG space where SVG_Y = -y:
      // Δx = -sin(θ), ΔSVG_Y = -cos(θ)
      const tanLen = 3.6;
      const dx = -Math.sin(angle);
      const dySvg = -Math.cos(angle);

      const tx1 = x - tanLen * dx;
      const ty1 = svgY - tanLen * dySvg;
      const tx2 = x + tanLen * dx;
      const ty2 = svgY + tanLen * dySvg;

      const tangent = document.getElementById('tangent-line');
      tangent.setAttribute('x1', tx1);
      tangent.setAttribute('y1', ty1);
      tangent.setAttribute('x2', tx2);
      tangent.setAttribute('y2', ty2);

      // Slope pill update
      const slopeText = document.getElementById('slope-text');
      const slopePill = document.getElementById('slope-pill');
      if (Math.abs(y) < 0.15) {
        slopePill.childNodes[0].nodeValue = `Slope at (${x.toFixed(1)}, ${y.toFixed(1)}): `;
        slopeText.textContent = 'm = undefined (vertical)';
      } else {
        const m = -x / y;
        slopePill.childNodes[0].nodeValue = `Slope at (${x.toFixed(1)}, ${y.toFixed(1)}): `;
        if (Math.abs(x - 3) < 0.15 && Math.abs(y - 4) < 0.15) {
          slopeText.textContent = 'm = −3/4';
        } else {
          slopeText.textContent = `m = ${m > 0 ? '+' : ''}${m.toFixed(2)}`;
        }
      }
    }

    updateTangent();

    const container = document.getElementById('circle-container');
    function handlePointer(e) {
      const rect = container.getBoundingClientRect();
      const cx = rect.left + rect.width / 2;
      const cy = rect.top + rect.height / 2;
      const px = e.clientX - cx;
      const py = -(e.clientY - cy);
      angle = Math.atan2(py, px);
      updateTangent();
    }

    container.addEventListener('pointerdown', (e) => {
      handlePointer(e);
      const onMove = (moveEv) => handlePointer(moveEv);
      const onUp = () => {
        window.removeEventListener('pointermove', onMove);
        window.removeEventListener('pointerup', onUp);
      };
      window.addEventListener('pointermove', onMove);
      window.addEventListener('pointerup', onUp);
    });
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>d/dx on x vs y Terms</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }

    body {
      background: #ffffff;
      color: #0f172a;
      padding: 16px 12px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100vh;
      box-sizing: border-box;
    }

    .container {
      width: 100%;
      max-width: 440px;
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 12px;
    }

    /* Controls bar */
    .controls {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 6px;
      width: 100%;
      flex-wrap: wrap;
    }

    .btn {
      background: #f1f5f9;
      border: 1px solid #cbd5e1;
      color: #1e293b;
      padding: 6px 12px;
      border-radius: 8px;
      font-size: 13px;
      font-weight: 600;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 4px;
      transition: background-color 0.2s, border-color 0.2s, color 0.2s, box-shadow 0.2s;
    }

    .btn:hover {
      background: #e2e8f0;
    }

    .btn.active {
      background: #0f2b5c;
      color: #ffffff;
      border-color: #0f2b5c;
      box-shadow: 0 2px 6px rgba(15, 43, 92, 0.25);
    }

    .play-btn {
      width: 32px;
      height: 32px;
      padding: 0;
      justify-content: center;
      border-radius: 50%;
      font-size: 13px;
      flex-shrink: 0;
    }

    /* Main visual stage */
    .stage {
      width: 100%;
      display: flex;
      flex-direction: column;
      gap: 12px;
    }

    /* Comparison rows */
    .row-card {
      background: #ffffff;
      border: 1.5px solid #cbd5e1;
      border-radius: 12px;
      padding: 12px 14px;
      display: flex;
      flex-direction: column;
      gap: 8px;
      transition: border-color 0.3s ease, box-shadow 0.3s ease, background-color 0.3s ease, opacity 0.3s ease;
      position: relative;
    }

    .row-card.highlight {
      border-color: #0284c7;
      background: #f0f9ff;
      box-shadow: 0 4px 14px rgba(2, 132, 199, 0.15);
    }

    .row-card.dimmed {
      background: #f8fafc;
      border-color: #e2e8f0;
      opacity: 0.85;
    }

    .row-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 12px;
      font-weight: 700;
      letter-spacing: 0.5px;
      text-transform: uppercase;
    }

    .tag-x {
      color: #334155;
    }

    .tag-y {
      color: #0369a1;
    }

    .badge {
      font-size: 11px;
      padding: 3px 8px;
      border-radius: 999px;
      font-weight: 600;
      background: #e2e8f0;
      color: #1e293b;
      border: 1px solid #cbd5e1;
    }

    .badge-chain {
      background: #e0f2fe;
      color: #0369a1;
      border: 1px solid #bae6fd;
    }

    /* Semantic Math Formatting */
    .math-row {
      display: flex;
      align-items: center;
      justify-content: center;
      gap: 8px;
      font-size: 19px;
      font-family: "Cambria Math", "Latin Modern Math", "STIX Two Math", "Times New Roman", serif;
      min-height: 48px;
      color: #0f172a;
    }

    .math-frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      font-size: 0.88em;
      line-height: 1.05;
      padding: 0 2px;
    }

    .math-num {
      border-bottom: 1.5px solid currentColor;
      padding-bottom: 1px;
    }

    .math-den {
      padding-top: 1px;
    }

    .math-var {
      font-style: italic;
      font-family: "Cambria Math", "Times New Roman", serif;
    }

    .eq-sign {
      font-weight: bold;
      color: #475569;
      font-family: system-ui, sans-serif;
      font-size: 17px;
    }

    .outer-part {
      color: #0f2b5c;
      font-weight: 700;
      padding: 3px 8px;
      border-radius: 6px;
      background: rgba(15, 43, 92, 0.08);
      border: 1px solid rgba(15, 43, 92, 0.2);
    }

    .inner-part {
      color: #0284c7;
      font-weight: 700;
      padding: 3px 8px;
      border-radius: 6px;
      background: rgba(2, 132, 199, 0.12);
      border: 1px solid rgba(2, 132, 199, 0.3);
    }

    .mult-dot {
      margin: 0 2px;
      font-size: 16px;
      color: #64748b;
    }

    /* Anatomy breakdown below y-row */
    .breakdown-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
      padding-top: 8px;
      border-top: 1px dashed #cbd5e1;
    }

    .breakdown-col {
      display: flex;
      flex-direction: column;
      align-items: center;
      text-align: center;
      font-size: 11px;
      gap: 2px;
    }

    .label-navy {
      color: #0f2b5c;
      font-weight: 700;
    }

    .label-blue {
      color: #0284c7;
      font-weight: 700;
    }

    .arrow-icon {
      font-size: 12px;
      line-height: 1;
      font-weight: 800;
    }

    .arrow-navy {
      color: #0f2b5c;
    }

    .arrow-blue {
      color: #0284c7;
    }

    .breakdown-desc {
      font-size: 10px;
      color: #475569;
      line-height: 1.2;
    }

    /* Explanation note */
    .note-box {
      font-size: 12px;
      line-height: 1.4;
      color: #334155;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 8px;
      padding: 8px 12px;
      width: 100%;
      text-align: center;
    }

    .note-box strong {
      color: #0f172a;
    }

    /* Stepper dots */
    .dots-container {
      display: flex;
      align-items: center;
      gap: 8px;
    }

    .dot {
      width: 10px;
      height: 10px;
      border-radius: 50%;
      background: #cbd5e1;
      cursor: pointer;
      border: none;
      transition: background-color 0.2s;
    }

    .dot.active {
      background: #0f2b5c;
    }

    @media (max-width: 360px) {
      .math-row {
        font-size: 17px;
        gap: 6px;
      }
      .outer-part, .inner-part {
        padding: 2px 6px;
      }
      .btn {
        padding: 5px 9px;
        font-size: 12px;
      }
    }
  </style>
</head>
<body>

  <div class="container">
    
    <div class="controls">
      <button class="btn play-btn" id="playBtn" onclick="togglePlay()" aria-label="Play or Pause Animation">⏸</button>
      <button class="btn active" id="tab0" onclick="goToStep(0)">Direct (<span class="math-var">x</span>)</button>
      <button class="btn" id="tab1" onclick="goToStep(1)">Chain (<span class="math-var">y</span>)</button>
      <button class="btn" id="tab2" onclick="goToStep(2)">Compare Both</button>
    </div>

    
    <div class="stage">
      
      
      <div class="row-card highlight" id="cardX">
        <div class="row-header">
          <span class="tag-x">x-variable (Standard)</span>
          <span class="badge">Direct Power Rule</span>
        </div>
        <div class="math-row">
          <span>
            <span class="math-frac">
              <span class="math-num">d</span>
              <span class="math-den">d<span class="math-var">x</span></span>
            </span>[<span class="math-var">x</span><sup>3</sup>]
          </span>
          <span class="eq-sign">=</span>
          <span class="outer-part">3<span class="math-var">x</span><sup>2</sup></span>
        </div>
      </div>

      
      <div class="row-card dimmed" id="cardY">
        <div class="row-header">
          <span class="tag-y">y-variable (<span class="math-var">y</span> is <span class="math-var">f(x)</span>)</span>
          <span class="badge badge-chain">Chain Rule</span>
        </div>
        <div class="math-row">
          <span>
            <span class="math-frac">
              <span class="math-num">d</span>
              <span class="math-den">d<span class="math-var">x</span></span>
            </span>[<span class="math-var">y</span><sup>3</sup>]
          </span>
          <span class="eq-sign">=</span>
          <span class="outer-part">3<span class="math-var">y</span><sup>2</sup></span>
          <span class="mult-dot">&sdot;</span>
          <span class="inner-part">
            <span class="math-frac">
              <span class="math-num">d<span class="math-var">y</span></span>
              <span class="math-den">d<span class="math-var">x</span></span>
            </span>
          </span>
        </div>

        
        <div class="breakdown-grid" id="breakdownGrid">
          <div class="breakdown-col">
            <span class="arrow-icon arrow-navy">&uarr;</span>
            <span class="label-navy">Outer Derivative</span>
            <span class="breakdown-desc">Differentiate as usual</span>
          </div>
          <div class="breakdown-col">
            <span class="arrow-icon arrow-blue">&uarr;</span>
            <span class="label-blue">Inner Factor</span>
            <span class="breakdown-desc">Multiply by <span class="math-var">dy/dx</span></span>
          </div>
        </div>
      </div>

    </div>

    
    <div class="note-box" id="noteBox">
      Differentiating with respect to <strong>x</strong> directly applies the power rule: <strong>d/dx [x³] = 3x²</strong>.
    </div>

    
    <div class="dots-container">
      <button class="dot active" onclick="goToStep(0)" aria-label="Step 1: Direct rule on x"></button>
      <button class="dot" onclick="goToStep(1)" aria-label="Step 2: Chain rule on y"></button>
      <button class="dot" onclick="goToStep(2)" aria-label="Step 3: Comparison"></button>
    </div>
  </div>

  <script>
    let currentStep = 0;
    let isPlaying = true;
    let timer = null;
    const STEP_INTERVAL = 4000; // 4 seconds per state, well above the 500ms stationary requirement

    const notes = [
      'Differentiating with respect to <strong>x</strong> directly applies the power rule: <strong>d/dx [x³] = 3x²</strong>.',
      'Because <strong>y is a function of x</strong>, applying the Chain Rule requires multiplying by the inner derivative <strong>dy/dx</strong>.',
      '<strong>Summary:</strong> Always append <strong>dy/dx</strong> when differentiating terms containing <strong>y</strong> with respect to <strong>x</strong>.'
    ];

    function updateView() {
      const cardX = document.getElementById('cardX');
      const cardY = document.getElementById('cardY');
      const noteBox = document.getElementById('noteBox');

      // Update tab buttons
      for (let i = 0; i < 3; i++) {
        const tab = document.getElementById('tab' + i);
        if (tab) tab.classList.toggle('active', currentStep === i);
      }

      // Update dots
      const dots = document.querySelectorAll('.dot');
      dots.forEach((dot, idx) => {
        dot.classList.toggle('active', currentStep === idx);
      });

      // Update explanatory note
      if (noteBox) {
        noteBox.innerHTML = notes[currentStep];
      }

      // Update card states with clear contrast
      if (currentStep === 0) {
        cardX.classList.remove('dimmed');
        cardX.classList.add('highlight');
        cardY.classList.add('dimmed');
        cardY.classList.remove('highlight');
      } else if (currentStep === 1) {
        cardX.classList.add('dimmed');
        cardX.classList.remove('highlight');
        cardY.classList.remove('dimmed');
        cardY.classList.add('highlight');
      } else {
        // Step 2: Compare both
        cardX.classList.remove('dimmed');
        cardX.classList.add('highlight');
        cardY.classList.remove('dimmed');
        cardY.classList.add('highlight');
      }
    }

    function goToStep(step) {
      currentStep = step;
      updateView();
      if (isPlaying) {
        restartTimer();
      }
    }

    function nextStep() {
      currentStep = (currentStep + 1) % 3;
      updateView();
    }

    function togglePlay() {
      isPlaying = !isPlaying;
      const playBtn = document.getElementById('playBtn');
      if (isPlaying) {
        playBtn.textContent = '⏸';
        restartTimer();
      } else {
        playBtn.textContent = '▶';
        clearInterval(timer);
      }
    }

    function restartTimer() {
      clearInterval(timer);
      timer = setInterval(nextStep, STEP_INTERVAL);
    }

    window.addEventListener('DOMContentLoaded', () => {
      updateView();
      timer = setInterval(nextStep, STEP_INTERVAL);
    });
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="1:1">
  <title>Tangent Equation: Constant vs Variable Slope</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8fafc;
      color: #1e293b;
      padding: 12px;
      line-height: 1.4;
    }

    .banner {
      background: #fff1f2;
      border-left: 4px solid #e11d48;
      border-radius: 8px;
      padding: 10px 12px;
      margin-bottom: 12px;
      display: flex;
      align-items: center;
      justify-content: space-between;
      gap: 8px;
      flex-wrap: wrap;
    }
    .banner-text {
      font-size: 13px;
      font-weight: 600;
      color: #9f1239;
    }
    .status-pill {
      background: #e11d48;
      color: #fff;
      font-size: 11px;
      font-weight: 700;
      padding: 3px 8px;
      border-radius: 999px;
      white-space: nowrap;
    }

    .grid-2 {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 12px;
      margin-bottom: 12px;
    }
    @media (max-width: 580px) {
      .grid-2 { grid-template-columns: 1fr; }
    }

    .method-card {
      background: #ffffff;
      border-radius: 12px;
      padding: 12px;
      border: 1px solid #e2e8f0;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }
    .card-correct {
      border-top: 4px solid #10b981;
    }
    .card-wrong {
      border-top: 4px solid #f43f5e;
    }

    .card-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
    }
    .card-title {
      font-size: 14px;
      font-weight: 700;
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .title-correct { color: #047857; }
    .title-wrong { color: #be123c; }

    .tag-badge {
      font-size: 10px;
      font-weight: 700;
      text-transform: uppercase;
      padding: 2px 6px;
      border-radius: 4px;
      letter-spacing: 0.5px;
    }
    .badge-correct { background: #d1fae5; color: #065f46; }
    .badge-wrong { background: #ffe4e6; color: #9f1239; }

    .math-box {
      background: #f8fafc;
      border: 1px solid #f1f5f9;
      border-radius: 6px;
      padding: 8px;
      font-size: 12px;
      display: flex;
      flex-direction: column;
      gap: 4px;
    }
    .math-row {
      display: flex;
      align-items: baseline;
      justify-content: space-between;
    }
    .math-label { color: #64748b; font-size: 11px; }
    .math-expr { font-family: "Courier New", monospace; font-weight: 600; }
    .hl-green {
      background: #10b981;
      color: #ffffff;
      padding: 1px 5px;
      border-radius: 3px;
    }
    .hl-red {
      background: #f43f5e;
      color: #ffffff;
      padding: 1px 5px;
      border-radius: 3px;
    }

    .plot-container {
      width: 100%;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 6px;
      overflow: hidden;
    }
    svg {
      display: block;
      width: 100%;
      height: auto;
    }

    .legend-row {
      display: flex;
      align-items: center;
      gap: 12px;
      font-size: 11px;
      color: #475569;
      padding-top: 2px;
    }
    .legend-item {
      display: flex;
      align-items: center;
      gap: 4px;
    }
    .legend-swatch {
      width: 12px;
      height: 3px;
      border-radius: 2px;
    }
    .swatch-curve { background: #3b82f6; }
    .swatch-tan { background: #10b981; }
    .swatch-cubic { background: #f43f5e; border-top: 1px dashed #f43f5e; }

    .slider-card {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 10px;
      padding: 10px 14px;
      display: flex;
      flex-direction: column;
      gap: 8px;
    }
    .slider-header {
      display: flex;
      justify-content: space-between;
      align-items: center;
      font-size: 12px;
      font-weight: 600;
      color: #334155;
    }
    .slider-wrap {
      display: flex;
      align-items: center;
      gap: 12px;
    }
    input[type=range] {
      flex: 1;
      accent-color: #2563eb;
      cursor: pointer;
    }
    .slider-feedback {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 8px;
      font-size: 12px;
    }
    .feedback-box {
      padding: 6px 8px;
      border-radius: 6px;
      font-family: "Courier New", monospace;
      font-weight: 600;
      text-align: center;
    }
    .fb-correct { background: #ecfdf5; color: #047857; border: 1px solid #a7f3d0; }
    .fb-wrong { background: #fff1f2; color: #be123c; border: 1px solid #fecdd3; }
  </style>
</head>
<body>

  
  <div class="banner">
    <div class="banner-text">
      ⚠️ Slope <i>m</i> must be evaluated to a <b>constant number</b> before point-slope form!
    </div>
    <div class="status-pill">Tangent = Line (Degree 1)</div>
  </div>

  
  <div class="grid-2">

    
    <div class="method-card card-correct">
      <div class="card-header">
        <span class="card-title title-correct">✓ Correct Method</span>
        <span class="tag-badge badge-correct">Linear Tangent</span>
      </div>

      <div class="math-box">
        <div class="math-row">
          <span class="math-label">1. Derivative:</span>
          <span class="math-expr">f'(x) = 3x² - 4</span>
        </div>
        <div class="math-row">
          <span class="math-label">2. At x = 2:</span>
          <span class="math-expr">m = f'(2) = <span class="hl-green">8</span></span>
        </div>
        <div class="math-row">
          <span class="math-label">3. Point-Slope:</span>
          <span class="math-expr">y - 1 = 8(x - 2) ⇒ <b>y = 8x - 15</b></span>
        </div>
      </div>

      <div class="plot-container">
        <svg id="plotCorrect" viewBox="0 0 260 150"></svg>
      </div>

      <div class="legend-row">
        <div class="legend-item">
          <span class="legend-swatch swatch-curve"></span>
          <span>f(x)</span>
        </div>
        <div class="legend-item">
          <span class="legend-swatch swatch-tan"></span>
          <span>Tangent y = 8x - 15</span>
        </div>
      </div>
    </div>

    
    <div class="method-card card-wrong">
      <div class="card-header">
        <span class="card-title title-wrong">✕ Common Mistake</span>
        <span class="tag-badge badge-wrong">Non-Linear Error</span>
      </div>

      <div class="math-box">
        <div class="math-row">
          <span class="math-label">1. Kept formula:</span>
          <span class="math-expr">m = <span class="hl-red">3x² - 4</span> (variable!)</span>
        </div>
        <div class="math-row">
          <span class="math-label">2. False Form:</span>
          <span class="math-expr">y - 1 = (3x² - 4)(x - 2)</span>
        </div>
        <div class="math-row">
          <span class="math-label">3. Result:</span>
          <span class="math-expr"><b>y = 3x³ - 6x² - 4x + 9</b> (Cubic!)</span>
        </div>
      </div>

      <div class="plot-container">
        <svg id="plotWrong" viewBox="0 0 260 150"></svg>
      </div>

      <div class="legend-row">
        <div class="legend-item">
          <span class="legend-swatch swatch-curve"></span>
          <span>f(x)</span>
        </div>
        <div class="legend-item">
          <span class="legend-swatch swatch-cubic"></span>
          <span>False Curve (Degree 3)</span>
        </div>
      </div>
    </div>

  </div>

  
  <div class="slider-card">
    <div class="slider-header">
      <span>Interactive Check: Test x-values</span>
      <span id="xValDisplay">x = 2.00</span>
    </div>
    <div class="slider-wrap">
      <input type="range" id="xSlider" min="0" max="3" step="0.05" value="2">
    </div>
    <div class="slider-feedback">
      <div class="feedback-box fb-correct" id="corrFeedback">
        Slope m = 8 (Constant Line)
      </div>
      <div class="feedback-box fb-wrong" id="wrongFeedback">
        3x² - 4 = 8.00 (Changes with x!)
      </div>
    </div>
  </div>

  <script>
    // Coordinate Mapper: [xMin, xMax] -> [0, 260], [yMin, yMax] -> [150, 0]
    const xMin = -0.5, xMax = 3.2;
    const yMin = -6, yMax = 12;

    function mapX(x) { return ((x - xMin) / (xMax - xMin)) * 260; }
    function mapY(y) { return 150 - ((y - yMin) / (yMax - yMin)) * 150; }

    function f(x) { return x * x * x - 4 * x + 1; }
    function tangentLine(x) { return 8 * x - 15; }
    function wrongCubic(x) { return 3 * Math.pow(x, 3) - 6 * Math.pow(x, 2) - 4 * x + 9; }

    function generatePath(fn, step = 0.05) {
      let d = "";
      for (let x = xMin; x <= xMax + 0.001; x += step) {
        const y = fn(x);
        const px = mapX(x).toFixed(1);
        const py = mapY(y).toFixed(1);
        d += (d === "" ? `M ${px} ${py}` : ` L ${px} ${py}`);
      }
      return d;
    }

    function createAxes() {
      const x0 = mapX(0).toFixed(1);
      const y0 = mapY(0).toFixed(1);
      let svg = `
        
        <line x1="0" y1="${y0}" x2="260" y2="${y0}" stroke="#e2e8f0" stroke-width="1.5" />
        <line x1="${x0}" y1="0" x2="${x0}" y2="150" stroke="#e2e8f0" stroke-width="1.5" />
      `;
      // Ticks at x = 1, 2, 3
      for (let tx = 1; tx <= 3; tx++) {
        const px = mapX(tx).toFixed(1);
        svg += `
          <line x1="${px}" y1="${mapY(-0.4)}" x2="${px}" y2="${mapY(0.4)}" stroke="#cbd5e1" stroke-width="1"/>
          <text x="${px}" y="${mapY(-1.5)}" font-size="9" fill="#94a3b8" text-anchor="middle">${tx}</text>
        `;
      }
      return svg;
    }

    // Render Correct Plot
    const svgCorr = document.getElementById("plotCorrect");
    const pX = mapX(2).toFixed(1);
    const pY = mapY(1).toFixed(1);

    svgCorr.innerHTML = `
      ${createAxes()}
      <path d="${generatePath(f)}" fill="none" stroke="#3b82f6" stroke-width="2.5" />
      <path d="${generatePath(tangentLine)}" fill="none" stroke="#10b981" stroke-width="2.5" />
      <circle cx="${pX}" cy="${pY}" r="4.5" fill="#10b981" stroke="#ffffff" stroke-width="1.5" />
      <text x="${parseFloat(pX) - 10}" y="${parseFloat(pY) - 10}" font-size="10" font-weight="700" fill="#047857">(2, 1)</text>
    `;

    // Render Wrong Plot
    const svgWrong = document.getElementById("plotWrong");
    svgWrong.innerHTML = `
      ${createAxes()}
      <path d="${generatePath(f)}" fill="none" stroke="#3b82f6" stroke-width="2.5" />
      <path d="${generatePath(wrongCubic)}" fill="none" stroke="#f43f5e" stroke-width="2" stroke-dasharray="4 3" />
      <circle cx="${pX}" cy="${pY}" r="4.5" fill="#f43f5e" stroke="#ffffff" stroke-width="1.5" />
      <text x="${parseFloat(pX) - 10}" y="${parseFloat(pY) - 10}" font-size="10" font-weight="700" fill="#be123c">(2, 1)</text>
    `;

    // Slider Interactive logic
    const slider = document.getElementById("xSlider");
    const xValDisplay = document.getElementById("xValDisplay");
    const corrFeedback = document.getElementById("corrFeedback");
    const wrongFeedback = document.getElementById("wrongFeedback");

    slider.addEventListener("input", (e) => {
      const x = parseFloat(e.target.value);
      xValDisplay.textContent = `x = ${x.toFixed(2)}`;
      
      const varSlope = (3 * x * x - 4).toFixed(2);
      
      corrFeedback.textContent = `Fixed Slope m = 8`;
      wrongFeedback.textContent = `3(${x.toFixed(2)})² - 4 = ${varSlope}`;

      if (Math.abs(x - 2.0) < 0.05) {
        wrongFeedback.style.background = "#fff1f2";
        wrongFeedback.style.borderColor = "#fecdd3";
      } else {
        wrongFeedback.style.background = "#ffe4e6";
        wrongFeedback.style.borderColor = "#fda4af";
      }
    });
  </script>
</body>
</html>

Finding Equations of Tangents to Curves

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <meta name="aspect-ratio" content="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
    }
    body {
      background: #ffffff;
      color: #1e2945;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 10px;
    }
    .graph-container {
      position: relative;
      width: 100%;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      overflow: hidden;
    }
    svg {
      display: block;
      width: 100%;
      height: auto;
    }
    .grid-line {
      stroke: #e2e8f0;
      stroke-width: 1;
      stroke-dasharray: 2,2;
    }
    .axis-line {
      stroke: #94a3b8;
      stroke-width: 1.5;
    }
    .axis-label {
      font-size: 11px;
      fill: #64748b;
      font-weight: 500;
    }
    .curve-path {
      fill: none;
      stroke: #2563eb;
      stroke-width: 3;
      stroke-linecap: round;
    }
    .tangent-line {
      stroke: #dc2626;
      stroke-width: 2.5;
      stroke-linecap: round;
      transition: all 0.05s ease-out;
    }
    .tangent-point {
      fill: #dc2626;
      stroke: #ffffff;
      stroke-width: 2.5;
    }
    .pulse-ring {
      fill: none;
      stroke: #dc2626;
      stroke-width: 2;
      opacity: 0.6;
      animation: pulse 2s infinite cubic-bezier(0.215, 0.61, 0.355, 1);
    }
    @keyframes pulse {
      0% { r: 5; opacity: 0.8; }
      100% { r: 16; opacity: 0; }
    }
    .point-label {
      font-size: 12px;
      font-weight: 700;
      fill: #1e2945;
      paint-order: stroke;
      stroke: #ffffff;
      stroke-width: 3px;
      stroke-linecap: round;
      stroke-linejoin: round;
    }
    .controls {
      display: flex;
      align-items: center;
      gap: 12px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 8px 14px;
    }
    .slider-label {
      font-size: 13px;
      font-weight: 600;
      color: #1e2945;
      white-space: nowrap;
    }
    input[type="range"] {
      flex: 1;
      accent-color: #2563eb;
      cursor: pointer;
      height: 6px;
    }
    .cards-row {
      display: grid;
      grid-template-columns: repeat(3, 1fr);
      gap: 8px;
    }
    .info-card {
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 8px 10px;
      display: flex;
      flex-direction: column;
      gap: 2px;
    }
    .card-title {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.5px;
      color: #64748b;
    }
    .card-math {
      font-size: 13px;
      font-weight: 700;
      color: #1e2945;
      font-family: "SF Pro Text", Consolas, Monaco, monospace;
      white-space: nowrap;
      overflow: hidden;
      text-overflow: ellipsis;
    }
    .card-sub {
      font-size: 11px;
      color: #dc2626;
      font-weight: 600;
      font-family: "SF Pro Text", Consolas, Monaco, monospace;
      white-space: nowrap;
      overflow: hidden;
      text-overflow: ellipsis;
    }
    @media (max-width: 440px) {
      .cards-row {
        grid-template-columns: 1fr;
      }
    }
  </style>
</head>
<body>

  <div class="graph-container">
    <svg id="plotSvg" viewBox="0 0 400 240">
      
      <g id="grid"></g>
      
      <line x1="30" y1="200" x2="385" y2="200" class="axis-line" />
      <line x1="60" y1="15" x2="60" y2="225" class="axis-line" />
      
      <text x="385" y="194" class="axis-label" font-weight="700">x</text>
      <text x="68" y="24" class="axis-label" font-weight="700">y</text>
      
      <path id="curve" class="curve-path" />
      
      <line id="tangent" class="tangent-line" />
      
      <circle id="pulseRing" class="pulse-ring" cx="0" cy="0" r="6" />
      <circle id="tangentDot" class="tangent-point" cx="0" cy="0" r="5" />
      <text id="pointLabel" class="point-label" x="0" y="0"></text>
    </svg>
  </div>

  <div class="controls">
    <span class="slider-label">Point x₀ = <span id="x0Value">1.0</span></span>
    <input type="range" id="x0Slider" min="0" max="3" step="0.05" value="1.0">
  </div>

  <div class="cards-row">
    <div class="info-card">
      <span class="card-title">1. Point</span>
      <span class="card-math">(x₀, f(x₀))</span>
      <span class="card-sub" id="ptReadout">(1.0, 3.5)</span>
    </div>
    <div class="info-card">
      <span class="card-title">2. Slope</span>
      <span class="card-math">m = f'(x₀)</span>
      <span class="card-sub" id="slopeReadout">m = 1.00</span>
    </div>
    <div class="info-card">
      <span class="card-title">3. Tangent</span>
      <span class="card-math">y - y₀ = m(x - x₀)</span>
      <span class="card-sub" id="eqReadout">y = 1.0x + 2.5</span>
    </div>
  </div>

  <script>
    // Mathematical setup: f(x) = -0.5*x^2 + 2*x + 2
    // f'(x) = -x + 2
    function f(x) {
      return -0.5 * x * x + 2 * x + 2;
    }
    function df(x) {
      return -x + 2;
    }

    // Viewport coordinate transform
    // Math bounds: x in [-0.5, 4.5], y in [-0.5, 4.8]
    const xMin = -0.5, xMax = 4.5;
    const yMin = -0.5, yMax = 4.8;
    const svgW = 400, svgH = 240;
    const padL = 60, padR = 25, padT = 20, padB = 40;
    const plotW = svgW - padL - padR;
    const plotH = svgH - padT - padB;

    function toSvgX(x) {
      return padL + ((x - 0) / (xMax - 0)) * plotW;
    }
    function toSvgY(y) {
      return (padT + plotH) - (y / yMax) * plotH;
    }

    // Render Grid
    const gridG = document.getElementById('grid');
    for (let x = 1; x <= 4; x++) {
      const sx = toSvgX(x);
      gridG.innerHTML += `<line x1="${sx}" y1="${padT}" x2="${sx}" y2="${toSvgY(0)}" class="grid-line" />`;
      gridG.innerHTML += `<text x="${sx}" y="${toSvgY(0) + 16}" class="axis-label" text-anchor="middle">${x}</text>`;
    }
    for (let y = 1; y <= 4; y++) {
      const sy = toSvgY(y);
      gridG.innerHTML += `<line x1="${toSvgX(0)}" y1="${sy}" x2="${padL + plotW}" y2="${sy}" class="grid-line" />`;
      gridG.innerHTML += `<text x="${toSvgX(0) - 8}" y="${sy + 4}" class="axis-label" text-anchor="end">${y}</text>`;
    }
    gridG.innerHTML += `<text x="${toSvgX(0) - 8}" y="${toSvgY(0) + 16}" class="axis-label" text-anchor="end">0</text>`;

    // Sample accurate parabola path
    const curve = document.getElementById('curve');
    let dStr = '';
    const numSamples = 60;
    for (let i = 0; i <= numSamples; i++) {
      const x = -0.2 + (i / numSamples) * 4.4;
      const y = f(x);
      const sx = toSvgX(x);
      const sy = toSvgY(y);
      dStr += (i === 0 ? `M ${sx.toFixed(1)} ${sy.toFixed(1)}` : ` L ${sx.toFixed(1)} ${sy.toFixed(1)}`);
    }
    curve.setAttribute('d', dStr);

    // Interactive state update
    const slider = document.getElementById('x0Slider');
    const x0Val = document.getElementById('x0Value');
    const tangent = document.getElementById('tangent');
    const tangentDot = document.getElementById('tangentDot');
    const pulseRing = document.getElementById('pulseRing');
    const pointLabel = document.getElementById('pointLabel');
    const ptReadout = document.getElementById('ptReadout');
    const slopeReadout = document.getElementById('slopeReadout');
    const eqReadout = document.getElementById('eqReadout');

    function update() {
      const x0 = parseFloat(slider.value);
      const y0 = f(x0);
      const m = df(x0);

      x0Val.textContent = x0.toFixed(1);

      // Tangent segment: span delta_x = +/- 1.8
      const x1 = Math.max(-0.3, x0 - 1.8);
      const y1 = y0 + m * (x1 - x0);
      const x2 = Math.min(4.3, x0 + 1.8);
      const y2 = y0 + m * (x2 - x0);

      tangent.setAttribute('x1', toSvgX(x1));
      tangent.setAttribute('y1', toSvgY(y1));
      tangent.setAttribute('x2', toSvgX(x2));
      tangent.setAttribute('y2', toSvgY(y2));

      const cx = toSvgX(x0);
      const cy = toSvgY(y0);
      tangentDot.setAttribute('cx', cx);
      tangentDot.setAttribute('cy', cy);
      pulseRing.setAttribute('cx', cx);
      pulseRing.setAttribute('cy', cy);

      // Point label position offset
      const labelYOffset = cy < 60 ? 20 : -12;
      pointLabel.setAttribute('x', cx);
      pointLabel.setAttribute('y', cy + labelYOffset);
      pointLabel.setAttribute('text-anchor', 'middle');
      pointLabel.textContent = `(${x0.toFixed(1)}, ${y0.toFixed(2)})`;

      // Readouts
      ptReadout.textContent = `(${x0.toFixed(1)}, ${y0.toFixed(2)})`;
      slopeReadout.textContent = `m = ${m >= 0 ? '+' : ''}${m.toFixed(2)}`;

      const c = y0 - m * x0;
      const cStr = c >= 0 ? `+ ${c.toFixed(2)}` : `- ${Math.abs(c).toFixed(2)}`;
      eqReadout.textContent = `y = ${m.toFixed(2)}x ${cStr}`;
    }

    slider.addEventListener('input', update);
    update();
  </script>
</body>
</html>

OGM.12.CAL.3

Aligned learning resources

Related diagrams