Unit 3: Composite, Implicit & Inverse Derivatives
1. The chain rule (free preview)
The chain rule handles compositions, functions inside functions, and it is the single most-used rule on the entire exam:
$\dfrac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$
In words: differentiate the outside (leaving the inside alone), then multiply by the derivative of the inside. Almost every hard-looking derivative on the AP exam is a chain rule problem wearing a costume.
Worked example
Differentiate $y = (2x + 1)^5$.
- Outside: fifth power. Its derivative, inside untouched: $5(2x+1)^4$.
- Inside: $2x + 1$. Its derivative: $2$.
- Multiply: $y' = 10(2x+1)^4$.
Common mistake: forgetting the inner derivative. If your answer to a composite derivative has no factor from the inside function, it is almost certainly wrong.
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The rest of Unit 3 is in the full course
Everything after the chain rule basics, including the topics students lose the most points on:
- Implicit differentiation, from circles to curve tangents
- Derivatives of inverse functions (the exam's favorite trap)
- Inverse trig derivatives: arcsin, arccos, arctan
- Higher-order derivatives
- 45 exam-style questions with verified step-by-step solutions
All 8 units, 300+ questions, 2 full-length exams. 7-day refund.