<!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, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: transparent;
      color: #1e293b;
      padding: 12px;
      display: flex;
      flex-direction: column;
      gap: 12px;
      min-height: 100%;
    }

    .grid {
      display: grid;
      grid-template-columns: repeat(auto-fit, minmax(260px, 1fr));
      gap: 12px;
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      border: 1.5px solid #e2e8f0;
      border-radius: 12px;
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      flex-direction: column;
      align-items: center;
      text-align: center;
      gap: 12px;
      transition: border-color 0.25s ease, box-shadow 0.25s ease, background-color 0.25s ease;
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      box-shadow: 0 0 0 2px rgba(2, 132, 199, 0.25);
      background: #f0f9ff;
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      box-shadow: 0 0 0 2px rgba(124, 58, 237, 0.2);
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    .badge {
      display: inline-flex;
      align-items: center;
      gap: 5px;
      font-size: 12px;
      font-weight: 700;
      letter-spacing: 0.03em;
      text-transform: uppercase;
      padding: 4px 10px;
      border-radius: 999px;
    }

    .badge-natural {
      background: #e0f2fe;
      color: #0369a1;
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    .badge-general {
      background: #f3e8ff;
      color: #7e22ce;
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      font-size: 17px;
      font-family: "Cambria Math", "KaTeX_Math", "Times New Roman", serif;
      min-height: 46px;
      display: flex;
      align-items: center;
      justify-content: center;
      flex-wrap: wrap;
      gap: 4px;
      color: #0f172a;
      line-height: 1.2;
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    .math-frac {
      display: inline-flex;
      flex-direction: column;
      vertical-align: middle;
      text-align: center;
      padding: 0 2px;
      font-size: 0.9em;
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    .math-frac .num {
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      font-style: italic;
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    .math-frac .den {
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    .math-var {
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      font-weight: 600;
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      border-color: #38bdf8;
      color: #0369a1;
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      border: 1.5px solid #e2e8f0;
      border-radius: 12px;
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      flex-direction: column;
      align-items: center;
      gap: 12px;
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    .bridge-title {
      font-size: 13px;
      font-weight: 700;
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      gap: 6px;
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    .bridge-steps {
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      align-items: center;
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      border-color: #eab308;
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      font-size: 14px;
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    .callout-text {
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      padding: 7px 16px;
      border-radius: 8px;
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      display: inline-flex;
      align-items: center;
      gap: 6px;
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      body {
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        font-size: 15px;
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        gap: 6px;
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      .pill-box {
        font-size: 12px;
        padding: 4px 8px;
      }
    }
  </style>
</head>
<body>

  
  <div class="grid">
    
    <div class="card-panel" id="cardNatural">
      <span class="badge badge-natural">Natural Base (e)</span>
      <div class="math-display">
        <span class="math-frac"><span class="num">d</span><span class="den">dx</span></span>
        <span>[<span class="math-var">e</span><sup><span class="math-var">kx</span></sup>]</span>
        <span class="math-op">=</span>
        <span><span class="math-var">k</span> · <span class="math-var">e</span><sup><span class="math-var">kx</span></sup></span>
      </div>
      <div class="factor-tag" id="factorNatural">
        Chain Factor: <span class="math-var">k</span>
      </div>
    </div>

    
    <div class="card-panel" id="cardGeneral">
      <span class="badge badge-general">General Base (a)</span>
      <div class="math-display" id="eqGeneralDisplay">
        <span class="math-frac"><span class="num">d</span><span class="den">dx</span></span>
        <span>[<span class="math-var" id="baseVar">a</span><sup><span class="math-var">kx</span></sup>]</span>
        <span class="math-op">=</span>
        <span id="generalFormulaResult"><span class="math-var">k</span> · ln(<span class="math-var">a</span>) · <span class="math-var">a</span><sup><span class="math-var">kx</span></sup></span>
      </div>
      <div class="factor-tag" id="factorGeneral">
        Extra Factor: ln(<span class="math-var">a</span>)
      </div>
    </div>
  </div>

  
  <div class="bridge-panel">
    <div class="bridge-title">
      <span>💡 Special Case Connection</span>
    </div>

    <div class="bridge-steps">
      <span class="pill-box" id="step1">If <span class="math-var">a</span> = <span class="math-var">e</span></span>
      <span class="step-arrow">→</span>
      <span class="pill-box" id="step2">ln(<span class="math-var">e</span>) = 1</span>
      <span class="step-arrow">→</span>
      <span class="pill-box" id="step3"><span class="math-var">k</span> · (1) · <span class="math-var">e</span><sup><span class="math-var">kx</span></sup> = <span class="math-var">k</span><span class="math-var">e</span><sup><span class="math-var">kx</span></sup></span>
    </div>

    <div class="callout-text" id="calloutExplanation">
      Since <strong>ln(e) = 1</strong>, the natural base derivative is simply a special case of the general base formula.
    </div>

    <button type="button" class="toggle-btn" id="toggleBtn">
      <span id="btnText">Substitute a = e in General Formula</span>
    </button>
  </div>

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    const btnText = document.getElementById('btnText');
    const cardNatural = document.getElementById('cardNatural');
    const cardGeneral = document.getElementById('cardGeneral');
    const baseVar = document.getElementById('baseVar');
    const generalFormulaResult = document.getElementById('generalFormulaResult');
    const factorGeneral = document.getElementById('factorGeneral');
    const factorNatural = document.getElementById('factorNatural');
    const step1 = document.getElementById('step1');
    const step2 = document.getElementById('step2');
    const step3 = document.getElementById('step3');
    const calloutExplanation = document.getElementById('calloutExplanation');

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    toggleBtn.addEventListener('click', function() {
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      toggleBtn.classList.toggle('active', isSubstituted);

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        btnText.textContent = 'Reset to Base a';
        cardNatural.classList.add('highlight-natural');
        cardGeneral.classList.add('highlight-general');

        baseVar.textContent = 'e';
        generalFormulaResult.innerHTML = '<span class="math-var">k</span> · <mark style="background:#fef08a; padding:1px 4px; border-radius:4px;">ln(e)</mark> · <span class="math-var">e</span><sup><span class="math-var">kx</span></sup> = <strong><span class="math-var">k</span> · <span class="math-var">e</span><sup><span class="math-var">kx</span></sup></strong>';
        factorGeneral.innerHTML = 'Extra Factor: ln(e) = <strong>1</strong>';
        factorGeneral.classList.add('highlight');
        factorNatural.classList.add('highlight');

        step1.classList.add('active');
        step2.classList.add('active');
        step3.classList.add('active');

        calloutExplanation.innerHTML = '<strong>Matched!</strong> When substituting <em>a = e</em>, ln(<em>e</em>) evaluates to 1, exactly yielding the natural base formula.';
      } else {
        btnText.textContent = 'Substitute a = e in General Formula';
        cardNatural.classList.remove('highlight-natural');
        cardGeneral.classList.remove('highlight-general');

        baseVar.textContent = 'a';
        generalFormulaResult.innerHTML = '<span class="math-var">k</span> · ln(<span class="math-var">a</span>) · <span class="math-var">a</span><sup><span class="math-var">kx</span></sup>';
        factorGeneral.innerHTML = 'Extra Factor: ln(<span class="math-var">a</span>)';
        factorGeneral.classList.remove('highlight');
        factorNatural.classList.remove('highlight');

        step1.classList.remove('active');
        step2.classList.remove('active');
        step3.classList.remove('active');

        calloutExplanation.innerHTML = 'Since <strong>ln(e) = 1</strong>, the natural base derivative is simply a special case of the general base formula.';
      }
    });
  </script>
</body>
</html>

Differentiating Exponential Functions with Natural and General Bases

<!DOCTYPE html>
<html lang="en">
<head>
  <meta charset="UTF-8">
  <meta name="viewport" content="width=device-width, initial-scale=1.0">
  <title>Exponential Derivatives at (0,1)</title>
  <style>
    * {
      box-sizing: border-box;
      margin: 0;
      padding: 0;
      user-select: none;
      -webkit-user-select: none;
    }

    body {
      font-family: system-ui, -apple-system, BlinkMacSystemFont, "Segoe UI", Roboto, sans-serif;
      background: #f8fafc;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      align-items: center;
      justify-content: space-between;
      width: 100%;
      min-height: 100%;
      padding: 10px 8px;
    }

    .toolbar {
      display: flex;
      gap: 6px;
      flex-wrap: wrap;
      justify-content: center;
      width: 100%;
      max-width: 480px;
      margin-bottom: 4px;
    }

    .btn {
      background: #ffffff;
      border: 1px solid #cbd5e1;
      border-radius: 16px;
      padding: 5px 11px;
      font-size: 12px;
      font-weight: 600;
      color: #475569;
      cursor: pointer;
      display: inline-flex;
      align-items: center;
      gap: 4px;
      transition: background 0.15s ease, color 0.15s ease, border-color 0.15s ease;
    }

    .btn:hover {
      background: #f1f5f9;
      color: #1e2945;
    }

    .btn:active {
      opacity: 0.85;
    }

    .btn.active {
      color: #ffffff;
      border-color: transparent;
    }

    .btn-2x.active { background: #64748b; }
    .btn-ex.active { background: #2563eb; }
    .btn-3x.active { background: #0f172a; }
    .btn-all.active { background: #3b82f6; }

    .graph-container {
      position: relative;
      width: 100%;
      max-width: 480px;
      aspect-ratio: 4 / 3;
      overflow: visible;
    }

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

    .badge-card {
      position: absolute;
      top: 6px;
      right: 6px;
      background: rgba(255, 255, 255, 0.92);
      border: 1px solid #e2e8f0;
      border-radius: 6px;
      padding: 5px 8px;
      font-size: 12px;
      font-weight: 700;
      color: #1e2945;
      display: flex;
      flex-direction: column;
      gap: 1px;
      pointer-events: none;
      backdrop-filter: blur(4px);
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    .badge-sub {
      font-size: 11px;
      font-weight: 600;
      color: #64748b;
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    .grid-line {
      stroke: #e2e8f0;
      stroke-width: 1;
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    .axis {
      stroke: #94a3b8;
      stroke-width: 1.5;
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    .axis-label {
      font-size: 11px;
      font-weight: 600;
      fill: #64748b;
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      stroke-linecap: round;
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      transition: opacity 0.25s ease, stroke-width 0.25s ease;
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      stroke: #f59e0b;
      stroke-width: 2;
      stroke-dasharray: 4 3;
      transition: all 0.25s ease;
    }

    .legend-row {
      display: flex;
      flex-wrap: wrap;
      gap: 10px 14px;
      justify-content: center;
      align-items: center;
      width: 100%;
      max-width: 480px;
      margin-top: 4px;
      font-size: 11.5px;
      font-weight: 600;
    }

    .legend-item {
      display: inline-flex;
      align-items: center;
      gap: 5px;
      white-space: nowrap;
    }

    .legend-indicator {
      width: 9px;
      height: 9px;
      border-radius: 50%;
      flex-shrink: 0;
    }

    sup {
      font-size: 0.75em;
      line-height: 0;
      position: relative;
      vertical-align: baseline;
      top: -0.4em;
    }

    @media (max-width: 380px) {
      .btn {
        padding: 4px 8px;
        font-size: 11px;
      }
      .legend-row {
        font-size: 10.5px;
        gap: 6px 10px;
      }
      .badge-card {
        padding: 4px 6px;
        font-size: 11px;
      }
      .badge-sub {
        font-size: 10px;
      }
    }
  </style>
</head>
<body>

  <div class="toolbar">
    <button class="btn btn-all active" onclick="setMode('all')">Compare All</button>
    <button class="btn btn-2x" onclick="setMode('2')"><em>y</em> = 2<sup>x</sup></button>
    <button class="btn btn-ex" onclick="setMode('e')"><em>y</em> = <em>e</em><sup>x</sup></button>
    <button class="btn btn-3x" onclick="setMode('3')"><em>y</em> = 3<sup>x</sup></button>
  </div>

  <div class="graph-container">
    <div class="badge-card" id="badgeCard">
      <span id="badgeTitle" style="color: #2563eb;">Natural Base (<em>e</em>)</span>
      <span class="badge-sub" id="badgeValue">Slope of <em>e</em><sup>x</sup> at (0,1) = 1.000</span>
    </div>

    <svg id="plot" viewBox="0 0 400 300">
      
      <g id="grid">
        <line class="grid-line" x1="40" y1="50" x2="360" y2="50" />
        <line class="grid-line" x1="40" y1="105" x2="360" y2="105" />
        <line class="grid-line" x1="40" y1="160" x2="360" y2="160" />
        <line class="grid-line" x1="40" y1="215" x2="360" y2="215" />
        <line class="grid-line" x1="40" y1="270" x2="360" y2="270" />
        <line class="grid-line" x1="110" y1="20" x2="110" y2="280" />
        <line class="grid-line" x1="180" y1="20" x2="180" y2="280" />
        <line class="grid-line" x1="250" y1="20" x2="250" y2="280" />
        <line class="grid-line" x1="320" y1="20" x2="320" y2="280" />
      </g>

      
      
      <line class="axis" x1="30" y1="270" x2="375" y2="270" />
      
      <line class="axis" x1="180" y1="15" x2="180" y2="285" />

      
      <text class="axis-label" x="110" y="284" text-anchor="middle">-1</text>
      <text class="axis-label" x="180" y="284" text-anchor="middle">0</text>
      <text class="axis-label" x="250" y="284" text-anchor="middle">1</text>
      <text class="axis-label" x="320" y="284" text-anchor="middle">2</text>
      <text class="axis-label" x="375" y="284" text-anchor="end">x</text>
      
      <text class="axis-label" x="172" y="219" text-anchor="end">1</text>
      <text class="axis-label" x="172" y="164" text-anchor="end">2</text>
      <text class="axis-label" x="172" y="109" text-anchor="end">3</text>
      <text class="axis-label" x="172" y="54" text-anchor="end">4</text>
      <text class="axis-label" x="172" y="20" text-anchor="end">y</text>

      
      <line id="tangent" class="tangent-line" x1="80" y1="270" x2="280" y2="160" />

      
      <path id="curve2" class="curve" stroke="#94a3b8" stroke-width="2.5" />
      <path id="curveE" class="curve" stroke="#2563eb" stroke-width="3" />
      <path id="curve3" class="curve" stroke="#1e2945" stroke-width="2.5" />

      
      <circle cx="180" cy="215" r="5" fill="#f59e0b" stroke="#ffffff" stroke-width="2" />
      <text x="190" y="212" font-size="12" font-weight="700" fill="#1e2945">(0, 1)</text>
    </svg>
  </div>

  <div class="legend-row">
    <div class="legend-item" style="color: #64748b;">
      <span class="legend-indicator" style="background: #94a3b8;"></span>
      <span>2<sup>x</sup> [ln(2) ≈ 0.693]</span>
    </div>
    <div class="legend-item" style="color: #2563eb;">
      <span class="legend-indicator" style="background: #2563eb;"></span>
      <span><em>e</em><sup>x</sup> [slope = 1.000]</span>
    </div>
    <div class="legend-item" style="color: #1e2945;">
      <span class="legend-indicator" style="background: #1e2945;"></span>
      <span>3<sup>x</sup> [ln(3) ≈ 1.099]</span>
    </div>
  </div>

  <script>
    const originX = 180;
    const originY = 270;
    const scaleX = 70;
    const scaleY = 55;

    function toSvg(x, y) {
      return {
        x: originX + x * scaleX,
        y: originY - y * scaleY
      };
    }

    function generatePath(base) {
      const points = [];
      const xMin = -2.0;
      const xMax = base === 2 ? 2.1 : (base === Math.E ? 1.55 : 1.35);
      const step = 0.05;

      for (let x = xMin; x <= xMax + 0.001; x += step) {
        const y = Math.pow(base, x);
        const pt = toSvg(x, y);
        points.push(`${pt.x.toFixed(1)},${pt.y.toFixed(1)}`);
      }
      return "M " + points.join(" L ");
    }

    // Set initial curve paths
    document.getElementById('curve2').setAttribute('d', generatePath(2));
    document.getElementById('curveE').setAttribute('d', generatePath(Math.E));
    document.getElementById('curve3').setAttribute('d', generatePath(3));

    function updateTangent(slope) {
      const x1 = -1.3;
      const x2 = 1.3;
      const pt1 = toSvg(x1, slope * x1 + 1);
      const pt2 = toSvg(x2, slope * x2 + 1);

      const tan = document.getElementById('tangent');
      tan.setAttribute('x1', pt1.x.toFixed(1));
      tan.setAttribute('y1', pt1.y.toFixed(1));
      tan.setAttribute('x2', pt2.x.toFixed(1));
      tan.setAttribute('y2', pt2.y.toFixed(1));
    }

    function setMode(mode) {
      const c2 = document.getElementById('curve2');
      const ce = document.getElementById('curveE');
      const c3 = document.getElementById('curve3');
      const badgeTitle = document.getElementById('badgeTitle');
      const badgeValue = document.getElementById('badgeValue');

      document.querySelectorAll('.btn').forEach(btn => btn.classList.remove('active'));

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        document.querySelector('.btn-all').classList.add('active');
        c2.style.opacity = '0.45';
        c2.style.strokeWidth = '2';
        ce.style.opacity = '1';
        ce.style.strokeWidth = '3.5';
        c3.style.opacity = '0.45';
        c3.style.strokeWidth = '2';

        updateTangent(1.0);
        badgeTitle.innerHTML = "Natural Base (<em>e</em>)";
        badgeTitle.style.color = "#2563eb";
        badgeValue.innerHTML = "Slope of <em>e</em><sup>x</sup> at (0,1) = 1.000";
      } else if (mode === '2') {
        document.querySelector('.btn-2x').classList.add('active');
        c2.style.opacity = '1';
        c2.style.strokeWidth = '3.5';
        ce.style.opacity = '0.2';
        ce.style.strokeWidth = '2';
        c3.style.opacity = '0.2';
        c3.style.strokeWidth = '2';

        const slope = Math.log(2);
        updateTangent(slope);
        badgeTitle.innerHTML = "Base 2: <em>y</em> = 2<sup>x</sup>";
        badgeTitle.style.color = "#64748b";
        badgeValue.innerHTML = `Slope = ln(2) ≈ ${slope.toFixed(3)} (&lt; 1)`;
      } else if (mode === 'e') {
        document.querySelector('.btn-ex').classList.add('active');
        c2.style.opacity = '0.2';
        c2.style.strokeWidth = '2';
        ce.style.opacity = '1';
        ce.style.strokeWidth = '3.5';
        c3.style.opacity = '0.2';
        c3.style.strokeWidth = '2';

        updateTangent(1.0);
        badgeTitle.innerHTML = "Base <em>e</em>: <em>y</em> = <em>e</em><sup>x</sup>";
        badgeTitle.style.color = "#2563eb";
        badgeValue.innerHTML = "Slope = ln(<em>e</em>) = 1.000";
      } else if (mode === '3') {
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        c2.style.opacity = '0.2';
        c2.style.strokeWidth = '2';
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  <div class="target-bar">
    <span class="target-label">Differentiate:</span>
    <span class="target-expr math">
      <span class="fraction"><span class="num"><i>d</i></span><span class="den"><i>dx</i></span></span>[<i>e</i><sup><i>kx</i></sup>]
    </span>
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        <span class="math-pill"><i>f</i>(<i>u</i>) = <i>e</i><sup><i>u</i></sup></span>
        <span class="arrow-icon">➔</span>
        <span class="math-pill"><i>f</i>&prime;(<i>u</i>) = <i>e</i><sup><i>u</i></sup></span>
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      <span class="card-title"><span class="badge-num">2</span> Inner Function</span>
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      <span class="card-title"><span class="badge-num">3</span> Chain Rule Product: <i>f</i>&prime;(<i>u</i>) &middot; <i>u</i>&prime;</span>
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        <span class="result-pill"><i>f</i>&prime;(<i>u</i>) &middot; <i>u</i>&prime;</span>
        <span>=</span>
        <span class="result-pill"><i>e</i><sup><i>kx</i></sup> &middot; <i>k</i></span>
        <span>=</span>
        <span class="result-pill final-highlight"><i>k</i><i>e</i><sup><i>kx</i></sup></span>
      </div>
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Differentiating Natural and General Logarithmic Functions

Differentiating Trigonometric Functions: Sine, Cosine, and Tangent

Differentiating Secant, Cosecant, and Cotangent Functions

The Chain Rule for Composite Power and Bracket Functions

The Chain Rule with Exponential and Logarithmic Functions

The Chain Rule with Trigonometric Functions

Using the Product Rule with Trigonometric and Exponential Functions

Using the Quotient Rule with Trigonometric and Exponential Functions

Differentiating Inverse Functions Using dx/dy

Using Inverse Trig Derivatives in Product and Quotient Rules

math

Advanced Differentiation Rules and Transcendental Functions

Differentiating Powers of x with Rational Exponents

Differentiating Polynomial Sums, Differences, and Algebraic Fractions

Differentiating Natural Exponential Functions e^(kx)

Differentiating General Exponential Functions a^(kx)

Differentiating the Natural Logarithm ln(x) and ln(kx)

Differentiating Sine, Cosine, and Tangent Functions

Differentiating Standard Functions

Implicit Differentiation for First Derivatives

Implicit Differentiation for Second Derivatives and Stationary Points

Differentiating Parametric Equations: First Derivative

Second Derivatives of Parametric Equations

Tangents and Rates of Change on Parametric Curves

Connected Rates of Change in Physical and Fluid Contexts

Related Rates with Angles, Shadows, and Moving Distances

Implicit and Parametric Differentiation and Rates of Change

The Product Rule for Multiplied Functions

The Quotient Rule for Fractional Functions

Differentiating Reciprocal Trigonometric Functions

Differentiating Inverse Trigonometric Functions

Advanced Differentiation Rules and Inverse Functions

Site Versus Situation in Geography

Major Landform Types: Mountains, Plateaus, and Plains

Fluvial Systems: Rivers, Watersheds, and Drainage Basins

Coastal Landforms: Estuaries, Deltas, and Barrier Islands

The Köppen-Geiger Climate Classification System

Latitude, Elevation, and Orographic Effects on Local Climate

Soil Horizons, Pedogenesis, and Agricultural Capacity

social-studies

Fundamentals of Physical Geography: Landforms, Climate, and Soils

Connected Rates of Change in Geometric Contexts

Constructing Differential Equations from Growth and Decay Contexts

Constructing Differential Equations in Kinematics and Mechanics

Connected Rates of Change and Differential Equations

The Physical Geography of the Great Lakes Basin

Physical Barriers and Natural Corridors in Human History

Physical Characteristics of Arid and Semi-Arid Regions

Physical Geography of Island and Archipelago Nations

Comparing Physical Landscapes Across World Regions

Regional Landscapes and World Geomorphology

Defining the Concept of Region in Geography

Formal (Uniform) Regions: Criteria and Boundaries

Functional (Nodal) Regions: Nodes and Hinterlands

Perceptual (Vernacular) Regions and Cultural Identity

The Cultural Landscape and Sequent Occupance

Linguistic Landscapes: Toponyms and Spatial Identity

Sacred Spaces and Religious Landscapes

Geographic Regions and Cultural Landscapes

Human Characteristics of the Great Lakes and Rust Belt

Regional Economic Specialization and Clustering

Regionalism, Nationalism, and Subnational Identities

Borderlands and Transnational Regions

Evaluating Dynamic Regional Boundaries Over Time

Regional Dynamics, Economics, and Borderlands

Cifras significativas y redondeo en cálculos métricos

Error absoluto, error relativo y error porcentual

Propagación de incertidumbres en mediciones indirectas

Tolerancia y especificaciones en diseño e ingeniería

Velocidad media como razón de cambio en cinemática

Aceleración media y variación de velocidad

Densidad media y presión en sistemas materiales

Caudal, flujo y otras razones de transferencia de materia

Procesos de aproximación sucesiva y método de exhaución

Rangos de variación y límites en mediciones continuas

Precisión de la medición, tolerancia y magnitudes derivadas

Elements of Culture: Traits, Complexes, and Systems

Cultural Hearths and the Spread of Early Civilizations

Global Distribution of Major Language Families

Dialects, Isoglosses, and Official Languages

Lingua Franca, Pidgins, and Creole Languages

Endangered Languages and Linguistic Revitalization

Spatial Diffusion of Universalizing Religions

Ethnic Religions: Hinduism, Judaism, and Indigenous Beliefs

Patterns of Culture, Language, and Religion

Folk Versus Popular Culture: Spatial Characteristics

Acculturation, Assimilation, and Syncretism

Cultural Homogenization Versus Cultural Heterogenization

Gender Roles and Spatial Inequality Across Cultures

Ethnic Enclaves and Multicultural Neighborhoods

Cultural Dynamics, Globalization, and Identity

Ecuación de la recta tangente y recta normal

Cinemática con derivadas: posición, velocidad y aceleración

Razones de cambio relacionadas: geometría

Razones de cambio relacionadas: física

Puntos críticos y criterio de la primera derivada

Concavidad, puntos de inflexión y criterio de la segunda derivada

Análisis de gráficas a partir de derivadas

Problemas de optimización geométrica: áreas y volúmenes

Problemas de optimización en economía y costos

Aplicaciones de la derivada y razones de cambio

Network Geography: Nodes, Links, and Connectivity

Intermodal Freight Transportation and Containerization

Global Maritime Chokepoints and Shipping Canals

Air Transportation: Hub-and-Spoke Networks

Rail and Highway Networks and Continental Integration

Global Transportation Networks and Spatial Mobility

Submarine Fiber-Optic Cables and Internet Backbones

The Global Digital Divide: Infrastructure and Access Gaps

Energy Transmission Grids and Pipeline Geography

Just-In-Time Manufacturing and Global Supply Chains

Telecommuting and the Decentralization of Work

Smart Cities and Urban Internet of Things (IoT)

Cyber Geography and Geopolitical Vulnerabilities

Digital Infrastructure, Telecommunications, and Cyber Geography

Measuring Population Density: Arithmetic, Physiological, and Agricultural

Analyzing Population Pyramids and Age-Sex Structures

The Demographic Transition Model (DTM): Stages 1 to 5

The Epidemiologic Transition Model

Malthusian Theory Versus Anti-Malthusian Perspectives

Migration Dynamics: Push and Pull Factors

Forced Migration: Refugees, Asylum Seekers, and IDPs

Voluntary Migration: Transnational, Step, and Chain Migration

Demographic Dynamics and Global Migration

Central Place Theory and Urban Hierarchies

Primate Cities Versus Rank-Size Rule

Classic Urban Models: Concentric, Sector, and Multiple-Nuclei

Suburbanization, Edge Cities, and Urban Sprawl

Gentrification, Urban Renewal, and Housing Segregation

Megacities and Informal Squatter Settlements

Urban Geography and Patterns of Settlement

The Concepts of State, Nation, Nation-State, and Stateless Nations

Types of Political Boundaries: Geometric, Physical, and Cultural

Boundary Evolution: Antecedent, Subsequent, Superimposed, and Relic

Boundary Disputes: Definitional, Locational, Operational, and Allocational

Law of the Sea (UNCLOS): Territorial Waters and Exclusive Economic Zones

Electoral Geography: Reapportionment, Redistricting, and Gerrymandering

Political Geography: States, Boundaries, and Maritime Law

Centripetal and Centrifugal Forces in Political Units

Devolution and Balkanization in Modern States

Supranationalism and the United Nations System

Regional Economic and Military Alliances: EU, NATO, and USMCA

Territoriality and Resource Conflicts in the South China Sea

Geopolitics of the Arctic: Melting Ice and Sovereignty

Transboundary Water Conflicts and Hydropolitics

Global Geopolitics: Alliances, Devolution, and Resource Conflicts

OGM.12.CAL.2

OGM.12.CAL.2

Aligned learning resources

Related diagrams