AP Calculus Chain Implicit Quiz – Free PDF

Contents

This ap calculus chain implicit quiz asks 10 multiple-choice questions on chain rule, implicit differentiation and inverse functions, and each answer comes with a short explanation.

Ten multiple-choice questions on the chain rule, implicit differentiation, inverse functions and second derivatives. Pick one answer for each, then read the explanation to see the common mistake.

AP Calculus ABCommon Core: AP Calculus AB Unit 3

🎯 Chain Rule, Implicit Differentiation and Inverse Functions – AP Calculus AB math quiz

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Question 1

Find \(\frac{d}{dx}(5x-2)^{4}\).

A \(20(5x-2)^{3}\)

B \(4(5x-2)^{3}\)

C \(5(5x-2)^{3}\)

D \(20(5x-2)^{4}\)

Question 2

Find \(\frac{d}{dx}\sin(x^{2})\).

A \(\cos(x^{2})\)

B \(2x\cos x\)

C \(2x\cos(x^{2})\)

D \(-2x\sin(x^{2})\)

Question 3

Find \(\frac{d}{dx}e^{3x^{2}}\).

A \(e^{3x^{2}}\)

B \(6x\,e^{3x^{2}}\)

C \(6x\,e^{3x}\)

D \(3x^{2}e^{3x^{2}}\)

Question 4

Use the figure to find the slope of the circle \(x^{2}+y^{2}=25\) at \((-3,4)\).

Circle of radius 5; the point (-3, 4) lies on the upper left part.

A \(\frac{3}{4}\)

B \(-\frac{3}{4}\)

C \(\frac{4}{3}\)

D \(-\frac{4}{3}\)

Question 5

Differentiating \(x^{2}+y^{3}=10\) implicitly with respect to \(x\) gives \(\frac{dy}{dx}\) equal to what?

A \(\frac{2x}{3y^{2}}\)

B \(-\frac{2x}{3y^{2}}\)

C \(-\frac{2x}{3y}\)

D \(-\frac{x}{y^{2}}\)

Question 6

Let \(f(x)=x^{3}+2x\), with graph and inverse shown. What is \((f^{-1})^{\prime}(3)\)?

Blue: f. Green: its reflection.

A \(5\)

B \(\frac{1}{29}\)

C \(\frac{1}{3}\)

D \(\frac{1}{5}\)

Question 7

Find \(\frac{d}{dx}\tan^{-1}(2x)\).

A \(\frac{1}{1+4x^{2}}\)

B \(\frac{2}{1+2x^{2}}\)

C \(\frac{2}{1+4x^{2}}\)

D \(\frac{2}{\sqrt{1-4x^{2}}}\)

Question 8

Let \(f(x)=x^{5}-2x^{3}\). What is \(f^{\prime\prime}(1)\)?

A \(-1\)

B \(48\)

C \(20\)

D \(8\)

Question 9

Find \(\frac{d^{2}}{dx^{2}}\cos(5x)\).

A \(-5\cos(5x)\)

B \(25\cos(5x)\)

C \(-25\sin(5x)\)

D \(-25\cos(5x)\)

Question 10

Let \(h(x)=f(g(x))\) with \(g(2)=5\), \(g^{\prime}(2)=3\), \(f^{\prime}(2)=4\) and \(f^{\prime}(5)=-2\). What is \(h^{\prime}(2)\)?

x → g → f

A \(-6\)

B \(12\)

C \(6\)

D \(1\)

🎉 Quiz complete!

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How does this quiz work?

Read each question, choose one answer among A, B, C and D, then check it: a short explanation appears after every answer. Your score out of 10 is shown at the end, and you can restart the quiz as often as you like.

What should you review before the quiz?

Reread the key facts of the lesson and the worked example. If a question is hard, go back to the matching part of the lesson, then try the quiz again. Write your work on scrap paper: most mistakes come from skipping a step, not from the idea itself.

Remember:

Chain rule: outer derivative evaluated at the inside, times the inner derivative.
Implicit differentiation: every y-term gets a factor of dy/dx from the chain rule.
Inverse functions: if f(a) = b, then (f⁻¹)′(b) = 1/f′(a).
The second derivative f″ is the derivative of f′ and measures how the slope changes.