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<h1>✅ Adventure 3 — Solutions</h1>
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These solutions show the “added pieces” and why the <code>dx²</code> and <code>dx³</code> terms vanish as <code>dx → 0</code>.
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Key idea: derivatives keep the terms proportional to <code>dx</code> and drop higher powers like <code>dx²</code>, <code>dx³</code>.
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<h2>Solution A — <code>f(x)=x²</code></h2>
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<img src="images/adventure_3_image_1_solution.png" alt="Square solution image">
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Added area pieces:
<br>vertical strip: <code>x·dx</code>
<br>horizontal strip: <code>x·dx</code>
<br>corner: <code>dx²</code>
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So <code>df(x) = x·dx + x·dx + dx² = 2x·dx + dx²</code>.
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Divide by <code>dx</code>:
<br><code>df(x)/dx = 2x + dx</code>.
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As <code>dx → 0</code>, the <code>dx</code> term vanishes, so
<br><strong><code>d(x²)/dx = 2x</code></strong>.
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<h2>Solution B — <code>f(x)=x³</code></h2>
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<img src="images/adventure_3_image_2_solution.png" alt="Cube solution image">
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Added volume pieces:
<br>3 face sheets: <code>3x²·dx</code>
<br>3 edge bars: <code>3x·dx²</code>
<br>1 corner cube: <code>dx³</code>
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<p>
So <code>df(x) = 3x²·dx + 3x·dx² + dx³</code>.
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Factor out <code>dx</code>:
<br><code>df(x) = (3x² + 3x·dx + dx²)·dx</code>.
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Divide by <code>dx</code>:
<br><code>df(x)/dx = 3x² + 3x·dx + dx²</code>.
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As <code>dx → 0</code>, the last two terms vanish, so
<br><strong><code>d(x³)/dx = 3x²</code></strong>.
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