Practice a College Board-style free response question on the chain rule, implicit differentiation, and inverse derivatives. Write your response, then reveal the model answer to see exactly what earns each point.
Let f be the function f(x) = (x² + 1)³, and let g be the inverse of the function h, where h(x) = x³ + x + 1.
A
Use implicit differentiation to find dy/dx for the curve x² + xy + y² = 7.
✓ Model answer
Differentiate both sides with respect to x, using the product rule on xy: 2x + [y + x(dy/dx)] + 2y(dy/dx) = 0. Group the dy/dx terms: (x + 2y)(dy/dx) = −(2x + y), so dy/dx = −(2x + y)/(x + 2y).
Why it scores: Differentiates every term correctly (product rule on xy, chain-rule dy/dx on y²), then solves for dy/dx. Forgetting the dy/dx on a y-term, or mishandling xy, loses points.
B
Find f′(x) for f(x) = (x² + 1)³, and evaluate f′(1).
✓ Model answer
By the chain rule, f′(x) = 3(x² + 1)²·(2x) = 6x(x² + 1)². At x = 1: f′(1) = 6(1)(1 + 1)² = 6·4 = 24.
Why it scores: Applies the chain rule with the correct outer derivative 3( )² and inner derivative 2x, then evaluates. Omitting the inner factor 2x is the most common error.
C
Let g be the inverse of h(x) = x³ + x + 1. Given that h(1) = 3, find g′(3).
✓ Model answer
Use g′(x) = 1/h′(g(x)). Since h(1) = 3, we have g(3) = 1. Compute h′(x) = 3x² + 1, so h′(1) = 3(1) + 1 = 4. Therefore g′(3) = 1/h′(1) = 1/4.
Why it scores: Correctly uses the inverse-derivative formula, identifies g(3) = 1 from h(1) = 3, computes h′(1) = 4, and reports 1/4. Forgetting the reciprocal, or evaluating h′ at the wrong point, loses credit.
How to score points on AP Calculus FRQs
Attach dy/dx to every y-term when differentiating implicitly, then solve cleanly.
Never drop the inner derivative in a chain-rule problem.
For inverse derivatives, find the matching point first, then take the reciprocal of h′ there.
Show your setup. Write the formula before plugging in numbers.
Answer the verb. "Find" needs the computation; "evaluate" needs a number.