The chain rule and implicit differentiation
Multiply by the inside derivative, and every y term picks up dy/dx.
The common mistake
Differentiating only the outside of a composite function — writing the derivative of (3x² + 1)⁴ as 4(3x² + 1)³ — or differentiating y² as 2y with no dy/dx.
Instead: Name the inside and outside first. Differentiate the outside with the inside left alone, then multiply by the derivative of the inside. When y depends on x, every y term you differentiate gets a dy/dx factor.
Worked example with Guide me
Find the derivative of (3x² + 1)⁴.
- Sugar · What's the inside, and what's the outside?Student · Inside is 3x² + 1, outside is something to the 4th.
- Sugar · Differentiate the outside, leaving the inside alone. What do you get?Student · 4(3x² + 1)³.
- Sugar · Now multiply by the inside's derivative. What is it?Student · 6x, so 4(3x² + 1)³ · 6x.
Answer: 24x(3x² + 1)³.
Try 3 practice questions
1. Differentiate (2x + 5)³.
2. Differentiate sin(4x).
3. If x² + y² = 25, find dy/dx.
Quick answers
- What is the most common mistake with the chain rule and implicit differentiation?
- Differentiating only the outside of a composite function — writing the derivative of (3x² + 1)⁴ as 4(3x² + 1)³ — or differentiating y² as 2y with no dy/dx.
- How do I get the chain rule and implicit differentiation right?
- Name the inside and outside first. Differentiate the outside with the inside left alone, then multiply by the derivative of the inside. When y depends on x, every y term you differentiate gets a dy/dx factor.
- Can you show a worked example?
- Find the derivative of (3x² + 1)⁴. 24x(3x² + 1)³.
Related concepts
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