<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Implicit Differentiation of a Circle</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      -webkit-tap-highlight-color: transparent;
    }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      min-height: 100vh;
      padding: 10px;
    }
    .main-layout {
      display: flex;
      flex-direction: row;
      flex-wrap: wrap;
      align-items: center;
      justify-content: center;
      gap: 16px;
      width: 100%;
      max-width: 660px;
    }
    .graph-container {
      position: relative;
      width: 270px;
      height: 270px;
      flex-shrink: 0;
      background: #ffffff;
      border-radius: 8px;
    }
    svg {
      width: 100%;
      height: 100%;
      display: block;
      overflow: visible;
    }
    .calc-panel {
      display: flex;
      flex-direction: column;
      gap: 8px;
      flex: 1;
      min-width: 260px;
      max-width: 340px;
    }
    .point-selector {
      display: flex;
      gap: 6px;
      width: 100%;
    }
    .pt-btn {
      flex: 1;
      padding: 6px 0;
      font-size: 13px;
      font-weight: 600;
      border: 1px solid #d1d5db;
      border-radius: 6px;
      background: #ffffff;
      color: #1e2945;
      cursor: pointer;
      box-shadow: 0 1px 2px rgba(0,0,0,0.05);
      transition: background-color 0.15s, color 0.15s, border-color 0.15s;
    }
    .pt-btn:hover {
      border-color: #2563eb;
      color: #2563eb;
    }
    .pt-btn.active {
      background: #2563eb;
      color: #ffffff;
      border-color: #2563eb;
    }
    .info-card {
      background: #f8fafc;
      border: 1px solid #e5e7eb;
      border-radius: 8px;
      padding: 8px 12px;
      display: flex;
      flex-direction: column;
      gap: 6px;
    }
    .card-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      font-size: 13px;
      min-height: 28px;
    }
    .badge {
      display: inline-flex;
      align-items: center;
      gap: 4px;
      font-weight: 600;
      font-size: 13px;
    }
    .badge-blue { color: #2563eb; }
    .badge-red { color: #ef4444; }
    .badge-purple { color: #7c3aed; }
    .badge-green { color: #16a34a; }

    /* Math typography styling */
    .math {
      font-family: "Cambria Math", "STIX Two Math", "Times New Roman", Georgia, serif;
      font-size: 15px;
      font-style: italic;
      color: #1e2945;
      display: inline-flex;
      align-items: center;
      gap: 2px;
    }
    .math .normal {
      font-style: normal;
    }
    .math-frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      font-size: 0.88em;
      line-height: 1;
      padding: 0 2px;
    }
    .math-num {
      border-bottom: 1.2px solid currentColor;
      padding-bottom: 2px;
    }
    .math-den {
      padding-top: 2px;
    }
    sub {
      font-size: 0.75em;
      line-height: 0;
      font-style: normal;
    }
    sup {
      font-size: 0.75em;
      line-height: 0;
      font-style: normal;
    }
  </style>
</head>
<body>

  <div class="main-layout">
    
    <div class="graph-container">
      <svg id="plot" viewBox="0 0 270 270">
        <defs>
          <pattern id="grid" width="22" height="22" patternUnits="userSpaceOnUse" x="5" y="5">
            <path d="M 22 0 L 0 0 0 22" fill="none" stroke="#e5e7eb" stroke-width="1"/>
          </pattern>
        </defs>

        
        <rect width="270" height="270" fill="url(#grid)" />

        
        <line x1="0" y1="135" x2="270" y2="135" stroke="#9ca3af" stroke-width="1.5" />
        <line x1="135" y1="0" x2="135" y2="270" stroke="#9ca3af" stroke-width="1.5" />

        
        <circle cx="135" cy="135" r="110" fill="rgba(37, 99, 235, 0.03)" stroke="#1e2945" stroke-width="2" />

        
        <path id="rightAngle" d="" fill="none" stroke="#1e2945" stroke-width="1.5" />

        
        <line id="radLine" x1="135" y1="135" x2="201" y2="47" stroke="#2563eb" stroke-width="2" stroke-dasharray="4,4" />

        
        <line id="tanLine" x1="113" y1="-19" x2="289" y2="113" stroke="#ef4444" stroke-width="2.5" stroke-linecap="round" />

        
        <circle cx="135" cy="135" r="3" fill="#9ca3af" />
        <text x="139" y="148" fill="#9ca3af" font-size="11" font-family="sans-serif">O</text>

        
        <circle id="ptDot" cx="201" cy="47" r="5.5" fill="#2563eb" stroke="#ffffff" stroke-width="2" />
        <text id="ptLabel" x="207" y="41" fill="#1e2945" font-size="12" font-weight="700" font-family="sans-serif">(3, 4)</text>
      </svg>
    </div>

    
    <div class="calc-panel">
      
      <div class="point-selector">
        <button class="pt-btn active" onclick="setPoint(3, 4, this)">(3, 4)</button>
        <button class="pt-btn" onclick="setPoint(-4, 3, this)">(-4, 3)</button>
        <button class="pt-btn" onclick="setPoint(4, -3, this)">(4, -3)</button>
        <button class="pt-btn" onclick="setPoint(-3, -4, this)">(-3, -4)</button>
      </div>

      
      <div class="info-card">
        <div class="card-row">
          <span class="badge badge-purple">Implicit Derivative</span>
          <span class="math">
            <span class="normal">2</span><i>x</i><span class="normal"> + 2</span><i>y</i>
            <span class="math-frac"><span class="math-num"><i>d</i><i>y</i></span><span class="math-den"><i>d</i><i>x</i></span></span>
            <span class="normal">= 0</span>
          </span>
        </div>
        <div class="card-row">
          <span style="font-size:13px; color:#4b5563; font-weight:500;">Slope Formula</span>
          <span class="math">
            <span class="math-frac"><span class="math-num"><i>d</i><i>y</i></span><span class="math-den"><i>d</i><i>x</i></span></span>
            <span class="normal">= −</span>
            <span class="math-frac"><span class="math-num"><i>x</i></span><span class="math-den"><i>y</i></span></span>
          </span>
        </div>
      </div>

      
      <div class="info-card">
        <div class="card-row">
          <span class="badge badge-red">Tangent Slope (<i>m</i><sub>1</sub>)</span>
          <span class="math" id="eqTan"></span>
        </div>
        <div class="card-row">
          <span class="badge badge-blue">Radial Slope (<i>m</i><sub>2</sub>)</span>
          <span class="math" id="eqRad"></span>
        </div>
      </div>

      
      <div class="info-card" style="border-left: 3px solid #16a34a; background: #f0fdf4;">
        <div class="card-row">
          <span class="badge badge-green">Perpendicular Check</span>
          <span class="math" id="eqPerp"></span>
        </div>
      </div>
    </div>
  </div>

  <script>
    const CX = 135, CY = 135, SCALE = 22;

    function renderFrac(num, den) {
      const isNegative = (num * den) < 0;
      const absN = Math.abs(num);
      const absD = Math.abs(den);

      if (absD === 1) {
        return `<span class="normal">${isNegative ? '−' : ''}${absN}</span>`;
      }
      return `${isNegative ? '<span class="normal">−</span>' : ''}<span class="math-frac"><span class="math-num">${absN}</span><span class="math-den">${absD}</span></span>`;
    }

    function setPoint(x, y, btn) {
      if (btn) {
        document.querySelectorAll('.pt-btn').forEach(b => b.classList.remove('active'));
        btn.classList.add('active');
      }

      const px = CX + x * SCALE;
      const py = CY - y * SCALE;

      // Update dot & label
      const ptDot = document.getElementById('ptDot');
      const ptLabel = document.getElementById('ptLabel');
      ptDot.setAttribute('cx', px);
      ptDot.setAttribute('cy', py);

      ptLabel.setAttribute('x', x >= 0 ? px + 7 : px - 46);
      ptLabel.setAttribute('y', y >= 0 ? py - 7 : py + 16);
      ptLabel.textContent = `(${x}, ${y})`;

      // Update radial line
      const radLine = document.getElementById('radLine');
      radLine.setAttribute('x2', px);
      radLine.setAttribute('y2', py);

      // Tangent line endpoints
      const len = 85;
      const ux = -y / 5;
      const uy = x / 5;
      const tanLine = document.getElementById('tanLine');
      tanLine.setAttribute('x1', px - ux * len);
      tanLine.setAttribute('y1', py + uy * len);
      tanLine.setAttribute('x2', px + ux * len);
      tanLine.setAttribute('y2', py - uy * len);

      // Right angle square indicator
      const s = 9;
      const rx = (x / 5);
      const ry = (-y / 5);
      const tx = (-y / 5);
      const ty = (-x / 5);

      const p1x = px - rx * s;
      const p1y = py - ry * s;
      const p2x = p1x - tx * s;
      const p2y = p1y - ty * s;
      const p3x = px - tx * s;
      const p3y = py - ty * s;

      const ra = document.getElementById('rightAngle');
      ra.setAttribute('d', `M ${p1x} ${p1y} L ${p2x} ${p2y} L ${p3x} ${p3y}`);

      // Tangent slope m1 = -x/y
      const tanFrac = renderFrac(-x, y);
      document.getElementById('eqTan').innerHTML = `<i>m</i><sub>1</sub> <span class="normal">=</span> ${tanFrac}`;

      // Radial slope m2 = y/x
      const radFrac = renderFrac(y, x);
      document.getElementById('eqRad').innerHTML = `<i>m</i><sub>2</sub> <span class="normal">=</span> ${radFrac}`;

      // Perpendicular check
      document.getElementById('eqPerp').innerHTML = `<i>m</i><sub>1</sub> <span class="normal">·</span> <i>m</i><sub>2</sub> <span class="normal">= −1</span>`;
    }

    window.addEventListener('DOMContentLoaded', () => {
      setPoint(3, 4, null);
    });
  </script>
</body>
</html>

Implicit Differentiation of Algebraic Curves

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Implicit Differentiation 4-Step Flow</title>
  <style>
    * { box-sizing: border-box; margin: 0; padding: 0; }
    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      padding: 12px 10px;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: flex-start;
      min-height: 100%;
    }

    /* Mathematical Typography */
    .math {
      font-family: "Cambria Math", "STIX Two Math", "Latin Modern Math", "Times New Roman", serif;
      font-size: 1.05em;
      letter-spacing: 0.02em;
    }
    .math-var {
      font-family: "Cambria Math", "Times New Roman", serif;
      font-style: italic;
      padding: 0 1px;
    }
    .math-sup {
      font-size: 0.75em;
      vertical-align: super;
      line-height: 0;
      margin-left: 0.5px;
    }
    .frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: -0.45em;
      text-align: center;
      font-size: 0.88em;
      line-height: 1;
      padding: 0 2px;
    }
    .frac .num {
      border-bottom: 1.2px solid currentColor;
      padding-bottom: 1px;
    }
    .frac .den {
      padding-top: 1.5px;
    }

    .header-bar {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      max-width: 580px;
      padding: 4px 6px 10px 6px;
      gap: 8px;
    }

    .eq-tag {
      background: #f8fafc;
      border: 1.5px solid #cbd5e1;
      padding: 5px 12px;
      border-radius: 999px;
      font-size: 13.5px;
      font-weight: 600;
      color: #0f172a;
      display: inline-flex;
      align-items: center;
      gap: 6px;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
    }
    .eq-tag .eq-label {
      font-size: 11px;
      text-transform: uppercase;
      letter-spacing: 0.05em;
      color: #64748b;
      font-family: system-ui, sans-serif;
      font-weight: 700;
    }

    .controls {
      display: flex;
      align-items: center;
      gap: 6px;
    }

    .ctrl-btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #1e2945;
      border-radius: 6px;
      padding: 5px 10px;
      font-size: 13px;
      cursor: pointer;
      font-weight: 600;
      line-height: 1;
      transition: background 0.15s, border-color 0.15s, box-shadow 0.15s;
      box-shadow: 0 1px 2px rgba(0,0,0,0.04);
    }
    .ctrl-btn:hover {
      background: #f1f5f9;
      border-color: #94a3b8;
    }
    .ctrl-btn:active {
      background: #e2e8f0;
    }

    .steps-container {
      display: flex;
      flex-direction: column;
      width: 100%;
      max-width: 580px;
      gap: 2px;
    }

    .step-item {
      display: grid;
      grid-template-columns: 28px 1fr auto;
      align-items: center;
      gap: 12px;
      padding: 9px 12px;
      border-radius: 10px;
      border: 1.5px solid #e2e8f0;
      background: #ffffff;
      cursor: pointer;
      transition: box-shadow 0.2s ease, border-color 0.2s ease, opacity 0.2s ease, background-color 0.2s ease;
      opacity: 0.6;
    }

    .step-item.active {
      opacity: 1;
      box-shadow: 0 3px 10px rgba(0, 0, 0, 0.07);
    }

    .step-num {
      width: 26px;
      height: 26px;
      border-radius: 50%;
      display: flex;
      align-items: center;
      justify-content: center;
      font-size: 12px;
      font-weight: 700;
      color: #ffffff;
      flex-shrink: 0;
    }

    .step-text {
      display: flex;
      flex-direction: column;
      gap: 2px;
      min-width: 0;
    }

    .step-title {
      font-size: 13px;
      font-weight: 700;
      color: #0f172a;
      line-height: 1.3;
    }

    .step-sub {
      font-size: 12px;
      font-weight: 500;
      color: #334155;
      line-height: 1.3;
    }

    .step-math {
      font-size: 13.5px;
      text-align: right;
      padding: 6px 10px;
      border-radius: 8px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      color: #0f172a;
      white-space: nowrap;
      font-weight: 500;
    }

    /* Step Color Themes */
    .step-1 { border-left: 5px solid #2563eb; }
    .step-1 .step-num { background: #2563eb; }
    .step-1.active { border-color: #93c5fd; border-left-color: #2563eb; background: #f0f6ff; }
    .step-1.active .step-math { background: #ffffff; border-color: #bfdbfe; }

    .step-2 { border-left: 5px solid #9333ea; }
    .step-2 .step-num { background: #9333ea; }
    .step-2.active { border-color: #d8b4fe; border-left-color: #9333ea; background: #faf5ff; }
    .step-2.active .step-math { background: #ffffff; border-color: #e9d5ff; }

    .step-3 { border-left: 5px solid #d97706; }
    .step-3 .step-num { background: #d97706; }
    .step-3.active { border-color: #fde68a; border-left-color: #d97706; background: #fffdf5; }
    .step-3.active .step-math { background: #ffffff; border-color: #fef08a; }

    .step-4 { border-left: 5px solid #16a34a; }
    .step-4 .step-num { background: #16a34a; }
    .step-4.active { border-color: #86efac; border-left-color: #16a34a; background: #f0fdf4; }
    .step-4.active .step-math { background: #ffffff; border-color: #bbf7d0; }

    .arrow-divider {
      display: flex;
      justify-content: center;
      align-items: center;
      height: 14px;
      color: #94a3b8;
      font-size: 13px;
      font-weight: bold;
      user-select: none;
    }

    @media (max-width: 520px) {
      .step-item {
        grid-template-columns: 24px 1fr;
        grid-template-rows: auto auto;
        row-gap: 6px;
        column-gap: 10px;
        padding: 8px 10px;
      }
      .step-num {
        width: 22px;
        height: 22px;
        font-size: 11px;
      }
      .step-title {
        font-size: 12.5px;
      }
      .step-sub {
        font-size: 11.5px;
      }
      .step-math {
        grid-column: 1 / span 2;
        text-align: left;
        font-size: 12.5px;
        padding: 5px 8px;
        overflow-x: auto;
      }
    }
  </style>
</head>
<body>

  <div class="header-bar">
    <div class="eq-tag">
      <span class="eq-label">Example:</span>
      <span class="math"><span class="math-var">x</span><span class="math-sup">3</span> + <span class="math-var">y</span><span class="math-sup">3</span> = 6<span class="math-var">x</span><span class="math-var">y</span></span>
    </div>
    <div class="controls">
      <button class="ctrl-btn" id="prevBtn" title="Previous step" aria-label="Previous step">◀</button>
      <button class="ctrl-btn" id="playBtn" title="Play/Pause" aria-label="Play or Pause">⏸</button>
      <button class="ctrl-btn" id="nextBtn" title="Next step" aria-label="Next step">▶</button>
    </div>
  </div>

  <div class="steps-container">
    
    <div class="step-item step-1 active" data-step="0">
      <div class="step-num">1</div>
      <div class="step-text">
        <div class="step-title">Differentiate w.r.t. <span class="math"><span class="math-var">x</span></span></div>
        <div class="step-sub">Attach <span class="math"><span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span></span> to <span class="math"><span class="math-var">y</span></span>-derivatives</div>
      </div>
      <div class="step-math math">
        3<span class="math-var">x</span><span class="math-sup">2</span> + 3<span class="math-var">y</span><span class="math-sup">2</span><span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span> = 6<span class="math-var">y</span> + 6<span class="math-var">x</span><span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span>
      </div>
    </div>

    <div class="arrow-divider">↓</div>

    
    <div class="step-item step-2" data-step="1">
      <div class="step-num">2</div>
      <div class="step-text">
        <div class="step-title">Expand &amp; Collect terms</div>
        <div class="step-sub"><span class="math"><span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span></span> terms on left, others on right</div>
      </div>
      <div class="step-math math">
        3<span class="math-var">y</span><span class="math-sup">2</span><span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span> − 6<span class="math-var">x</span><span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span> = 6<span class="math-var">y</span> − 3<span class="math-var">x</span><span class="math-sup">2</span>
      </div>
    </div>

    <div class="arrow-divider">↓</div>

    
    <div class="step-item step-3" data-step="2">
      <div class="step-num">3</div>
      <div class="step-text">
        <div class="step-title">Factor out <span class="math"><span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span></span></div>
        <div class="step-sub">Group common derivative factor</div>
      </div>
      <div class="step-math math">
        (3<span class="math-var">y</span><span class="math-sup">2</span> − 6<span class="math-var">x</span>)<span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span> = 6<span class="math-var">y</span> − 3<span class="math-var">x</span><span class="math-sup">2</span>
      </div>
    </div>

    <div class="arrow-divider">↓</div>

    
    <div class="step-item step-4" data-step="3">
      <div class="step-num">4</div>
      <div class="step-text">
        <div class="step-title">Divide to isolate <span class="math"><span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span></span></div>
        <div class="step-sub">Divide &amp; simplify fraction</div>
      </div>
      <div class="step-math math">
        <span class="frac"><span class="num">d<span class="math-var">y</span></span><span class="den">d<span class="math-var">x</span></span></span> = <span class="frac"><span class="num">2<span class="math-var">y</span> − <span class="math-var">x</span><span class="math-sup">2</span></span><span class="den"><span class="math-var">y</span><span class="math-sup">2</span> − 2<span class="math-var">x</span></span></span>
      </div>
    </div>
  </div>

  <script>
    document.addEventListener("DOMContentLoaded", () => {
      const stepItems = document.querySelectorAll('.step-item');
      const prevBtn = document.getElementById('prevBtn');
      const nextBtn = document.getElementById('nextBtn');
      const playBtn = document.getElementById('playBtn');

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

      function showStep(idx) {
        currentStep = (idx + stepItems.length) % stepItems.length;
        stepItems.forEach((el, i) => {
          el.classList.toggle('active', i === currentStep);
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      function startTimer() {
        if (timer) clearInterval(timer);
        timer = setInterval(() => {
          showStep(currentStep + 1);
        }, 3200);
      }

      function stopTimer() {
        if (timer) {
          clearInterval(timer);
          timer = null;
        }
      }

      stepItems.forEach((item, idx) => {
        item.addEventListener('click', () => {
          showStep(idx);
          if (isPlaying) startTimer();
        });
      });

      prevBtn.addEventListener('click', () => {
        showStep(currentStep - 1);
        if (isPlaying) startTimer();
      });

      nextBtn.addEventListener('click', () => {
        showStep(currentStep + 1);
        if (isPlaying) startTimer();
      });

      playBtn.addEventListener('click', () => {
        isPlaying = !isPlaying;
        playBtn.textContent = isPlaying ? '⏸' : '▶';
        if (isPlaying) {
          startTimer();
        } else {
          stopTimer();
        }
      });

      startTimer();
    });
  </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="4:3">
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }

    body {
      font-family: -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, Helvetica, Arial, sans-serif;
      background: transparent;
      color: #0f172a;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: center;
      padding: 8px;
      user-select: none;
      -webkit-user-select: none;
    }

    .comparison-grid {
      display: grid;
      grid-template-columns: 1fr 1fr;
      gap: 16px;
      width: 100%;
      max-width: 680px;
    }

    @media (max-width: 520px) {
      .comparison-grid {
        grid-template-columns: 1fr;
        gap: 12px;
      }
    }

    .panel {
      display: flex;
      flex-direction: column;
      align-items: center;
      background: #ffffff;
      border: 1px solid #e2e8f0;
      border-radius: 12px;
      padding: 12px;
    }

    .panel-header {
      display: flex;
      flex-direction: column;
      align-items: center;
      gap: 4px;
      margin-bottom: 8px;
      width: 100%;
    }

    .title-row {
      display: flex;
      align-items: center;
      justify-content: space-between;
      width: 100%;
    }

    .title {
      font-size: 15px;
      font-weight: 700;
      color: #0f172a;
    }

    .badge {
      font-size: 11px;
      font-weight: 600;
      padding: 2px 8px;
      border-radius: 12px;
      letter-spacing: 0.2px;
    }

    .badge-pass {
      background: #eff6ff;
      color: #2563eb;
      border: 1px solid #bfdbfe;
    }

    .badge-fail {
      background: #fff1f2;
      color: #e11d48;
      border: 1px solid #fecdd3;
    }

    .formula {
      font-family: ui-monospace, SFMono-Regular, Menlo, Monaco, Consolas, monospace;
      font-size: 14px;
      font-weight: 700;
      color: #2563eb;
      background: #f8fafc;
      padding: 2px 10px;
      border-radius: 6px;
      border: 1px solid #e2e8f0;
    }

    .svg-container {
      width: 100%;
      aspect-ratio: 1 / 1;
      max-width: 260px;
      position: relative;
    }

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

    .caption {
      font-size: 12px;
      color: #475569;
      text-align: center;
      margin-top: 8px;
      line-height: 1.3;
      font-weight: 500;
    }

    .slider-row {
      display: flex;
      align-items: center;
      gap: 8px;
      margin-top: 8px;
      width: 100%;
      max-width: 440px;
      padding: 6px 12px;
      background: #f8fafc;
      border: 1px solid #e2e8f0;
      border-radius: 20px;
    }

    .slider-label {
      font-size: 12px;
      font-weight: 700;
      color: #475569;
      font-family: ui-monospace, monospace;
      white-space: nowrap;
    }

    input[type="range"] {
      flex: 1;
      accent-color: #2563eb;
      cursor: pointer;
      height: 4px;
    }

    .grid-line {
      stroke: #f1f5f9;
      stroke-width: 1;
    }

    .axis-line {
      stroke: #cbd5e1;
      stroke-width: 1.5;
    }

    .tick-label {
      font-size: 9px;
      fill: #94a3b8;
      font-family: ui-monospace, monospace;
      text-anchor: middle;
      dominant-baseline: middle;
    }
  </style>
</head>
<body>

  <div class="comparison-grid">
    
    <div class="panel">
      <div class="panel-header">
        <div class="title-row">
          <span class="title">Explicit Function</span>
          <span class="badge badge-pass">Passes VLT (1:1)</span>
        </div>
        <span class="formula">y = x² − 4</span>
      </div>

      <div class="svg-container">
        <svg id="svgExplicit" viewBox="0 0 200 200"></svg>
      </div>

      <div class="caption">
        Each input <span style="font-family:monospace;font-weight:700;">x</span> gives <strong>exactly 1</strong> output <span style="font-family:monospace;font-weight:700;">y</span>.
      </div>
    </div>

    
    <div class="panel">
      <div class="panel-header">
        <div class="title-row">
          <span class="title">Implicit Relation</span>
          <span class="badge badge-fail">Fails Global VLT</span>
        </div>
        <span class="formula">x² + y² = 25</span>
      </div>

      <div class="svg-container">
        <svg id="svgImplicit" viewBox="0 0 200 200"></svg>
      </div>

      <div class="caption">
        <strong>Multiple outputs</strong> globally, but local smooth slopes still exist.
      </div>
    </div>
  </div>

  
  <div class="slider-row">
    <span class="slider-label">Vertical Test Probe: x = <span id="probeVal">2.0</span></span>
    <input type="range" id="probeSlider" min="-4.5" max="4.5" step="0.1" value="2.0">
  </div>

  <script>
    const svgExp = document.getElementById('svgExplicit');
    const svgImp = document.getElementById('svgImplicit');
    const probeSlider = document.getElementById('probeSlider');
    const probeVal = document.getElementById('probeVal');

    // Coordinate mapping parameters (viewBox 0 0 200 200)
    // Scale: 1 unit = 16px, origin = (100, 100) -> range [-5.5, 5.5]
    const ORIGIN = 100;
    const SCALE = 16;

    function toSvgX(x) { return ORIGIN + x * SCALE; }
    function toSvgY(y) { return ORIGIN - y * SCALE; }

    function buildBaseGrid(svg) {
      let g = '';
      // Grid lines
      for (let i = -5; i <= 5; i++) {
        const sx = toSvgX(i);
        const sy = toSvgY(i);
        g += `<line class="grid-line" x1="${sx}" y1="10" x2="${sx}" y2="190" />`;
        g += `<line class="grid-line" x1="10" y1="${sy}" x2="190" y2="${sy}" />`;
      }
      // Axes
      g += `<line class="axis-line" x1="10" y1="${ORIGIN}" x2="190" y2="${ORIGIN}" />`;
      g += `<line class="axis-line" x1="${ORIGIN}" y1="10" x2="${ORIGIN}" y2="190" />`;
      // Arrowheads/labels
      g += `<text class="tick-label" x="192" y="${ORIGIN + 10}">x</text>`;
      g += `<text class="tick-label" x="${ORIGIN - 10}" y="12">y</text>`;
      // Axis markers
      [-4, -2, 2, 4].forEach(val => {
        g += `<text class="tick-label" x="${toSvgX(val)}" y="${ORIGIN + 11}">${val}</text>`;
        g += `<text class="tick-label" x="${ORIGIN - 9}" y="${toSvgY(val)}">${val}</text>`;
      });
      return g;
    }

    // Generate accurate path for y = x^2 - 4
    function generateParabolaPath() {
      let d = '';
      const minX = -3.1;
      const maxX = 3.1;
      const steps = 60;
      for (let i = 0; i <= steps; i++) {
        const x = minX + (maxX - minX) * (i / steps);
        const y = x * x - 4;
        const sx = toSvgX(x);
        const sy = toSvgY(y);
        d += (i === 0 ? 'M' : 'L') + `${sx.toFixed(1)},${sy.toFixed(1)}`;
      }
      return d;
    }

    const baseExpSvg = buildBaseGrid(svgExp) + `
      <path d="${generateParabolaPath()}" fill="none" stroke="#2563eb" stroke-width="2.5" stroke-linecap="round" />
    `;

    const circleR = 5 * SCALE; // 80px
    const baseImpSvg = buildBaseGrid(svgImp) + `
      <circle cx="${ORIGIN}" cy="${ORIGIN}" r="${circleR}" fill="none" stroke="#2563eb" stroke-width="2.5" />
    `;

    function render(probeX) {
      probeVal.textContent = probeX.toFixed(1);
      const probeSvgX = toSvgX(probeX);

      // --- Explicit Graph Update ---
      const expY = probeX * probeX - 4;
      const expSvgY = toSvgY(expY);
      const inExpBounds = expSvgY >= 8 && expSvgY <= 192;

      let expDynamic = `
        <line x1="${probeSvgX}" y1="12" x2="${probeSvgX}" y2="188" stroke="#64748b" stroke-width="1.5" stroke-dasharray="4,4" />
      `;

      if (inExpBounds) {
        expDynamic += `
          <circle cx="${probeSvgX}" cy="${expSvgY}" r="4.5" fill="#2563eb" stroke="#ffffff" stroke-width="1.5" />
          <rect x="${Math.min(Math.max(probeSvgX - 48, 12), 104)}" y="${Math.max(expSvgY - 24, 14)}" width="86" height="18" rx="4" fill="#0f172a" opacity="0.85" />
          <text x="${Math.min(Math.max(probeSvgX - 5, 55), 147)}" y="${Math.max(expSvgY - 12, 26)}" fill="#ffffff" font-size="10" font-weight="600" font-family="monospace" text-anchor="middle">
            (${probeX.toFixed(1)}, ${expY.toFixed(1)})
          </text>
        `;
      }
      svgExp.innerHTML = baseExpSvg + expDynamic;

      // --- Implicit Graph Update ---
      const rSq = 25;
      const xSq = probeX * probeX;
      let impDynamic = `
        <line x1="${probeSvgX}" y1="12" x2="${probeSvgX}" y2="188" stroke="#64748b" stroke-width="1.5" stroke-dasharray="4,4" />
      `;

      if (xSq <= rSq) {
        const yTop = Math.sqrt(rSq - xSq);
        const yBot = -yTop;
        const svgYTop = toSvgY(yTop);
        const svgYBot = toSvgY(yBot);

        // Top Intersection & Tangent
        impDynamic += `
          <circle cx="${probeSvgX}" cy="${svgYTop}" r="4.5" fill="#059669" stroke="#ffffff" stroke-width="1.5" />
        `;

        if (yTop > 0.05) {
          // Tangent line at (x, yTop): slope m = -x / yTop
          const m = -probeX / yTop;
          const dx = 1.4;
          const x1 = probeX - dx;
          const y1 = yTop - m * dx;
          const x2 = probeX + dx;
          const y2 = yTop + m * dx;
          impDynamic += `
            <line x1="${toSvgX(x1)}" y1="${toSvgY(y1)}" x2="${toSvgX(x2)}" y2="${toSvgY(y2)}" stroke="#059669" stroke-width="2" stroke-linecap="round" />
          `;
        }

        // Bottom Intersection
        if (Math.abs(yTop - yBot) > 0.3) {
          impDynamic += `
            <circle cx="${probeSvgX}" cy="${svgYBot}" r="4.5" fill="#e11d48" stroke="#ffffff" stroke-width="1.5" />
            <rect x="${Math.min(Math.max(probeSvgX - 44, 12), 112)}" y="${Math.min(svgYBot + 8, 170)}" width="80" height="18" rx="4" fill="#e11d48" opacity="0.9" />
            <text x="${Math.min(Math.max(probeSvgX - 4, 52), 152)}" y="${Math.min(svgYBot + 20, 182)}" fill="#ffffff" font-size="9.5" font-weight="700" font-family="monospace" text-anchor="middle">
              2nd: y = ${yBot.toFixed(1)}
            </text>
          `;
        }

        // Top Label
        impDynamic += `
          <rect x="${Math.min(Math.max(probeSvgX - 44, 12), 112)}" y="${Math.max(svgYTop - 24, 12)}" width="80" height="18" rx="4" fill="#059669" opacity="0.9" />
          <text x="${Math.min(Math.max(probeSvgX - 4, 52), 152)}" y="${Math.max(svgYTop - 12, 24)}" fill="#ffffff" font-size="9.5" font-weight="700" font-family="monospace" text-anchor="middle">
            1st: y = +${yTop.toFixed(1)}
          </text>
        `;
      }

      svgImp.innerHTML = baseImpSvg + impDynamic;
    }

    probeSlider.addEventListener('input', (e) => {
      render(parseFloat(e.target.value));
    });

    // Initial render at x = 3.0 to match classic 3-4-5 implicit triangle
    probeSlider.value = 3.0;
    render(3.0);
  </script>
</body>
</html>

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Implicit vs Explicit Differentiation</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      color: #1e2945;
      background: transparent;
      padding: 14px;
      display: flex;
      flex-direction: column;
      gap: 14px;
      min-height: 100%;
      justify-content: center;
    }

    .diff-box {
      border-radius: 12px;
      padding: 14px 16px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      position: relative;
      border: 1.5px solid transparent;
      transition: opacity 0.35s ease, border-color 0.35s ease, box-shadow 0.35s ease;
    }

    .box-x {
      background: #eff6ff;
      border-color: #bfdbfe;
    }

    .box-y {
      background: #fef2f2;
      border-color: #fecaca;
    }

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

    .badge {
      font-size: 11px;
      font-weight: 700;
      text-transform: uppercase;
      letter-spacing: 0.05em;
      padding: 3px 8px;
      border-radius: 6px;
    }

    .badge-x {
      background: #dbeafe;
      color: #1d4ed8;
    }

    .badge-y {
      background: #fee2e2;
      color: #b91c1c;
    }

    .sub-tag {
      font-size: 12px;
      font-weight: 600;
      color: #64748b;
    }

    .math-row {
      display: flex;
      align-items: center;
      flex-wrap: wrap;
      gap: 8px 12px;
      font-size: 1.15rem;
      line-height: 1.2;
    }

    /* Standardized HTML/CSS Math Rendering */
    .math {
      font-family: "Cambria Math", "STIX Two Math", "Times New Roman", serif, Georgia;
      display: inline-flex;
      align-items: center;
      white-space: nowrap;
    }

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

    .math .num,
    .math .op,
    .math .text {
      font-style: normal;
    }

    .math .op {
      margin: 0 3px;
    }

    .math sup {
      font-style: normal;
      font-size: 0.72em;
      position: relative;
      top: -0.45em;
      line-height: 0;
      vertical-align: baseline;
      margin-left: 1px;
    }

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

    .frac-top {
      border-bottom: 1.5px solid currentColor;
      padding-bottom: 1px;
    }

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

    .arrow {
      color: #94a3b8;
      font-size: 15px;
      font-weight: bold;
      user-select: none;
    }

    .factor-container {
      display: inline-flex;
      align-items: center;
      gap: 6px;
      flex-wrap: wrap;
    }

    .chain-pill {
      background: #ffffff;
      border: 1.5px solid #ef4444;
      border-radius: 6px;
      padding: 2px 7px;
      box-shadow: 0 0 8px rgba(239, 68, 68, 0.25);
      display: inline-flex;
      align-items: center;
    }

    .chain-callout {
      display: inline-flex;
      align-items: center;
      gap: 4px;
      background: #dc2626;
      color: #ffffff;
      font-size: 11px;
      font-weight: 700;
      padding: 4px 9px;
      border-radius: 12px;
      white-space: nowrap;
      box-shadow: 0 0 10px rgba(220, 38, 38, 0.45);
      animation: pulseGlow 2.5s infinite ease-in-out;
    }

    @keyframes pulseGlow {
      0%, 100% {
        box-shadow: 0 0 8px rgba(220, 38, 38, 0.35);
      }
      50% {
        box-shadow: 0 0 14px rgba(220, 38, 38, 0.65);
      }
    }

    .controls {
      display: flex;
      justify-content: center;
      gap: 8px;
      margin-top: 2px;
    }

    .btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      color: #334155;
      padding: 6px 14px;
      border-radius: 6px;
      font-size: 12px;
      font-weight: 600;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 6px;
      transition: background 0.3s ease, border-color 0.3s ease, color 0.3s ease, box-shadow 0.3s ease;
    }

    .btn:hover {
      background: #f8fafc;
      border-color: #94a3b8;
    }

    .btn.active {
      background: #1e2945;
      color: #ffffff;
      border-color: #1e2945;
    }

    .dimmed {
      opacity: 0.3;
    }

    @media (max-width: 480px) {
      body {
        padding: 8px;
        gap: 10px;
      }
      .diff-box {
        padding: 10px 12px;
        gap: 8px;
      }
      .math-row {
        font-size: 1.02rem;
        gap: 6px 8px;
      }
      .chain-callout {
        font-size: 10px;
        padding: 3px 7px;
      }
      .btn {
        padding: 5px 10px;
        font-size: 11px;
      }
    }
  </style>
</head>
<body>

  
  <div class="diff-box box-x" id="boxX">
    <div class="box-header">
      <span class="badge badge-x">Matches Variable (x)</span>
      <span class="sub-tag">Standard Power Rule</span>
    </div>
    <div class="math-row">
      <div class="math">
        <span class="frac"><span class="frac-top">d</span><span class="frac-bottom">d<span class="var">x</span></span></span>[<span class="var">x</span><sup>3</sup>]
      </div>
      <span class="arrow">➔</span>
      <div class="math">
        <span class="num">3</span><span class="var">x</span><sup>2</sup>
      </div>
    </div>
  </div>

  
  <div class="diff-box box-y" id="boxY">
    <div class="box-header">
      <span class="badge badge-y">Composite Function y(x)</span>
      <span class="sub-tag">Requires Chain Rule</span>
    </div>
    <div class="math-row" id="mathRowY">
      
      <div class="math">
        <span class="frac"><span class="frac-top">d</span><span class="frac-bottom">d<span class="var">x</span></span></span>[<span class="var">y</span><sup>3</sup>]
      </div>

      <span class="arrow">➔</span>

      
      <div class="math">
        <span class="frac"><span class="frac-top">d</span><span class="frac-bottom">d<span class="var">y</span></span></span>[<span class="var">y</span><sup>3</sup>]<span class="op">·</span><span class="frac"><span class="frac-top">d<span class="var">y</span></span><span class="frac-bottom">d<span class="var">x</span></span></span>
      </div>

      <span class="arrow">➔</span>

      
      <div class="math">
        <span class="num">3</span><span class="var">y</span><sup>2</sup><span class="op">·</span>
      </div>

      <div class="factor-container" id="factorY">
        <div class="chain-pill">
          <span class="math"><span class="frac"><span class="frac-top">d<span class="var">y</span></span><span class="frac-bottom">d<span class="var">x</span></span></span></span>
        </div>
        <span class="chain-callout">✦ Chain Rule Factor (<span class="math"><span class="frac"><span class="frac-top">d<span class="var">y</span></span><span class="frac-bottom">d<span class="var">x</span></span></span></span>)</span>
      </div>
    </div>
  </div>

  
  <div class="controls">
    <button class="btn active" id="btnBoth" onclick="setView('both')">Overview</button>
    <button class="btn" id="btnX" onclick="setView('x')">Focus on x</button>
    <button class="btn" id="btnY" onclick="setView('y')">Focus on y</button>
  </div>

  <script>
    function setView(mode) {
      const boxX = document.getElementById('boxX');
      const boxY = document.getElementById('boxY');
      const btnBoth = document.getElementById('btnBoth');
      const btnX = document.getElementById('btnX');
      const btnY = document.getElementById('btnY');

      btnBoth.classList.remove('active');
      btnX.classList.remove('active');
      btnY.classList.remove('active');

      if (mode === 'both') {
        btnBoth.classList.add('active');
        boxX.classList.remove('dimmed');
        boxY.classList.remove('dimmed');
      } else if (mode === 'x') {
        btnX.classList.add('active');
        boxX.classList.remove('dimmed');
        boxY.classList.add('dimmed');
      } else if (mode === 'y') {
        btnY.classList.add('active');
        boxX.classList.add('dimmed');
        boxY.classList.remove('dimmed');
      }
    }
  </script>
</body>
</html>

OGM.13.CAL.3

Aligned learning resources

Related diagrams