AP Calculus AB The Chain Rule Worksheet (PDF)

AP Calculus AB math worksheet: The Chain Rule – Maths PDF

Cover of the book AP Calculus AB Math Worksheets with Answers (PDF)

Download as a PDFThe AP Calculus AB worksheets book with answersPrintable PDFSee the book ›

Sommaire

This chain rule worksheet for AP Calculus AB covers the chain rule (Common Core AP CALC AB 3.1), with a free PDF to print.

These 22 problems practice the chain rule for composite functions, including powers, trig, exponential and logarithmic layers (AP Calculus AB, topics 3.1). The early problems build the basic skill, and each one uses a different function or situation so that no two feel alike.

Later problems use three layers, the chain rule inside products and quotients, tables of values and real situations such as cooling coffee and a pendulum. Work in pencil, show every step, and check the answer key only after a full attempt.

AP Calculus AB · Common Core State Standards: AP CALC AB 3.1

Chain rule worksheet: exercises 1 to 10

Exercise 1 – Power of a linear function

Differentiate \(f(x)=(3x+2)^{5}\).

Exercise 2 – Power of a quadratic

Differentiate \(f(x)=(x^{2}-4x)^{3}\).

Exercise 3 – Square root of a linear function

Differentiate \(f(x)=\sqrt{5x-1}\).

Exercise 4 – Reciprocal power

Differentiate \(f(x)=\frac{1}{(2x+3)^{2}}\).

Exercise 5 – Sine of a multiple

Differentiate \(f(x)=\sin 4x\).

Exercise 6 – Cosine of a power

Differentiate \(f(x)=\cos(x^{2})\).

Exercise 7 – Exponential with a multiple

Differentiate \(f(x)=e^{3x}\).

Exercise 8 – Bell curve

The bell curve e^(−x²) and its derivative.

Differentiate \(f(x)=e^{-x^{2}}\) and explain why the derivative is zero at \(x=0\).

Exercise 9 – Log of a linear function

Differentiate \(f(x)=\ln(5x+2)\).

Exercise 10 – Log of a quadratic

Differentiate \(f(x)=\ln(x^{2}+1)\).

Exercises 11 and up

Exercise 11 – Sine squared

Differentiate \(f(x)=\sin^{2}x\).

Exercise 12 – Tangent of a multiple

Differentiate \(f(x)=\tan 3x\).

Exercise 13 – Sine of an exponential

Differentiate \(f(x)=\sin(e^{x})\) and \(g(x)=e^{\sin x}\).

Exercise 14 – Three layers

Differentiate \(f(x)=\cos^{3}(2x)\).

Exercise 15 – Chain inside a product

Differentiate \(f(x)=x^{2}\sqrt{x^{2}+1}\).

Exercise 16 – Chain inside a quotient

Differentiate \(f(x)=\frac{e^{2x}}{x+1}\).

Exercise 17 – Composite from a table

Let \(h(x)=f(g(x))\). Given \(g(1)=3\), \(g^{\prime}(1)=2\), \(f(3)=5\) and \(f^{\prime}(3)=-4\), find \(h^{\prime}(1)\).

Exercise 18 – Tangent line with the chain rule

The tangent at x = 1.

Find the tangent line to \(y=(x^{2}+1)^{3}\) at \(x=1\).

Exercise 19 – Horizontal tangents

The graph of f(x) = x e^(−x²/2).

Find where \(f(x)=xe^{-x^{2}/2}\) has a horizontal tangent.

Exercise 20 – Cooling coffee

Cooling coffee: temperature in °F after t minutes.

The temperature of Priya’s coffee in °F is \(T(t)=70+110e^{-0.05t}\) after \(t\) minutes. Find \(T^{\prime}(10)\) and interpret it.

Exercise 21 – Pendulum

Pendulum angle in radians.

A pendulum’s angle in radians is \(\theta(t)=0.3\cos 2t\) after \(t\) seconds. Find the angular velocity \(\theta^{\prime}(t)\) and evaluate it at \(t=\frac{\pi}{4}\).

Exercise 22 – Nested radicals

Differentiate \(f(x)=\sqrt{x+\sqrt{x}}\).

Key facts behind this chain rule worksheet

Key facts

Chain rule: dy/dx = f′(g(x)) · g′(x), or dy/du times du/dx.
Outer derivative first, inner function unchanged, then multiply by the inner derivative.
d/dx e^(g(x)) = e^(g(x)) g′(x) and d/dx ln(g(x)) = g′(x)/g(x).
Functions with three layers need the chain rule twice.

How to use this chain rule worksheet

Work through the exercises in order: they go from simple to challenging. Write every step on paper, then compare with the solutions page and redo any exercise you missed the next day.

All AP Calculus AB chapters: AP Calculus AB math.

Voter.. post

Free printable PDF