AP Calculus AB The Chain Rule Worksheet Solutions (PDF)
This chain rule answers for AP Calculus AB covers the chain rule (Common Core AP CALC AB 3.1), with a free PDF to print.
This answer key gives a full worked solution for every problem in the chain rule worksheet. Compare each step, not only the final answer, because AP graders award credit for a correct setup and clear justification. Final answers are in bold. Redo any problem you missed before moving on to the test.
AP Calculus AB · Common Core State Standards: AP CALC AB 3.1
Chain rule answers: exercises 1 to 10
Exercise 1 – Power of a linear function
Outer function: \(u^{5}\). Inner function: \(u=3x+2\).
Chain rule: \(f^{\prime}(x)=5(3x+2)^{4}\cdot 3\).
Final answer: \(f^{\prime}(x)=15 \left(3 x + 2\right)^{4}\).
Exercise 2 – Power of a quadratic
Outer function: \(u^{3}\). Inner function: \(u=x^{2}-4x\).
Chain rule: \(f^{\prime}(x)=3(x^{2}-4x)^{2}\cdot(2x-4)\).
Final answer: \(f^{\prime}(x)=3 \left(2 x – 4\right) \left(x^{2} – 4 x\right)^{2}\).
Exercise 3 – Square root of a linear function
Outer function: \(\sqrt{u}\). Inner function: \(u=5x-1\).
Chain rule: \(f^{\prime}(x)=\frac{1}{2\sqrt{5x-1}}\cdot 5\).
Final answer: \(f^{\prime}(x)=\frac{5}{2 \sqrt{5 x – 1}}\).
Exercise 4 – Reciprocal power
Rewrite: \(f(x)=(2x+3)^{-2}\).
Outer function: \(u^{-2}\). Inner function: \(u=2x+3\).
Chain rule: \(f^{\prime}(x)=-2(2x+3)^{-3}\cdot 2\).
Final answer: \(f^{\prime}(x)=- \frac{4}{\left(2 x + 3\right)^{3}}\).
Exercise 5 – Sine of a multiple
Outer function: \(\sin u\). Inner function: \(u=4x\).
Chain rule: \(f^{\prime}(x)=\cos(4x)\cdot 4\).
Final answer: \(f^{\prime}(x)=4 \cos{\left(4 x \right)}\).
Exercise 6 – Cosine of a power
Outer function: \(\cos u\). Inner function: \(u=x^{2}\).
Chain rule: \(f^{\prime}(x)=-\sin(x^{2})\cdot 2x\).
Final answer: \(f^{\prime}(x)=- 2 x \sin{\left(x^{2} \right)}\).
Exercise 7 – Exponential with a multiple
Outer function: \(e^{u}\). Inner function: \(u=3x\).
Chain rule: \(f^{\prime}(x)=e^{3x}\cdot 3\).
Final answer: \(f^{\prime}(x)=3 e^{3 x}\).
Exercise 8 – Bell curve
Outer function: \(e^{u}\). Inner function: \(u=-x^{2}\).
Chain rule: \(f^{\prime}(x)=e^{-x^{2}}\cdot(-2x)\).
Final answer: \(f^{\prime}(x)=- 2 x e^{- x^{2}}\).
At \(x=0\) the factor \(-2x\) is \(0\), which matches the peak of the bell curve.
Exercise 9 – Log of a linear function
Outer function: \(\ln u\). Inner function: \(u=5x+2\).
Chain rule: \(f^{\prime}(x)=\frac{1}{5x+2}\cdot 5\).
Final answer: \(f^{\prime}(x)=\frac{5}{5 x + 2}\).
Exercise 10 – Log of a quadratic
Outer function: \(\ln u\). Inner function: \(u=x^{2}+1\).
Chain rule: \(f^{\prime}(x)=\frac{1}{x^{2}+1}\cdot 2x\).
Final answer: \(f^{\prime}(x)=\frac{2 x}{x^{2} + 1}\).
Exercises 11 and up
Exercise 11 – Sine squared
Outer function: \(u^{2}\). Inner function: \(u=\sin x\).
Chain rule: \(f^{\prime}(x)=2\sin x\cdot\cos x\).
Final answer: \(f^{\prime}(x)=2 \sin{\left(x \right)} \cos{\left(x \right)}\).
This equals \(\sin 2x\) by a double-angle identity.
Exercise 12 – Tangent of a multiple
Outer function: \(\tan u\). Inner function: \(u=3x\).
Chain rule: \(f^{\prime}(x)=\sec^{2}(3x)\cdot 3\).
Final answer: \(f^{\prime}(x)=3 \sec^{2}{\left(3 x \right)}\).
Exercise 13 – Sine of an exponential
For \(f\): outer \(\sin u\), inner \(e^{x}\), so \(f^{\prime}(x)=\cos(e^{x})\cdot e^{x}\).
For \(g\): outer \(e^{u}\), inner \(\sin x\), so \(g^{\prime}(x)=e^{\sin x}\cos x\).
Final answer: \(f^{\prime}(x)=e^{x}\cos(e^{x})\) and \(g^{\prime}(x)=e^{\sin x}\cos x\).
Exercise 14 – Three layers
Layers: cube, cosine, then \(2x\). Work from the outside:
\(f^{\prime}(x)=3\cos^{2}(2x)\cdot(-\sin 2x)\cdot 2\).
Final answer: \(-6\cos^{2}(2x)\sin(2x)\).
Exercise 15 – Chain inside a product
Product rule with \(u=x^{2}\) and \(v=\sqrt{x^{2}+1}\). By the chain rule, \(v^{\prime}=\frac{2x}{2\sqrt{x^{2}+1}}=\frac{x}{\sqrt{x^{2}+1}}\).
\(f^{\prime}(x)=2x\sqrt{x^{2}+1}+\frac{x^{3}}{\sqrt{x^{2}+1}}=\frac{2x(x^{2}+1)+x^{3}}{\sqrt{x^{2}+1}}\).
Final answer: \(\frac{3x^{3}+2x}{\sqrt{x^{2}+1}}\).
Exercise 16 – Chain inside a quotient
Quotient rule with \(u=e^{2x}\), \(u^{\prime}=2e^{2x}\) (chain rule), \(v=x+1\), \(v^{\prime}=1\).
\(f^{\prime}(x)=\frac{2e^{2x}(x+1)-e^{2x}}{(x+1)^{2}}=\frac{e^{2x}(2x+1)}{(x+1)^{2}}\).
Final answer: \(\frac{e^{2x}(2x+1)}{(x+1)^{2}}\).
Exercise 17 – Composite from a table
Chain rule: \(h^{\prime}(1)=f^{\prime}(g(1))\cdot g^{\prime}(1)=f^{\prime}(3)\cdot 2=(-4)(2)=-8\).
Final answer: \(-8\).
Exercise 18 – Tangent line with the chain rule
\(y(1)=2^{3}=8\). \(y^{\prime}=3(x^{2}+1)^{2}\cdot 2x=6x(x^{2}+1)^{2}\), so \(y^{\prime}(1)=6\cdot 4=24\).
Tangent: \(y-8=24(x-1)\). Final answer: \(y=24x-16\).
Exercise 19 – Horizontal tangents
Product and chain rules: \(f^{\prime}(x)=e^{-x^{2}/2}+x\cdot e^{-x^{2}/2}\cdot(-x)=e^{-x^{2}/2}(1-x^{2})\).
Since the exponential is never zero, set \(1-x^{2}=0\).
Final answer: \(x=1\) and \(x=-1\).
Exercise 20 – Cooling coffee
\(T^{\prime}(t)=110\cdot e^{-0.05t}\cdot(-0.05)=-5.5e^{-0.05t}\).
\(T^{\prime}(10)=-5.5e^{-0.5}\approx -3.34\).
Final answer: \(T^{\prime}(10)\approx -3.34\text{ °F/min}\): after 10 minutes the coffee is cooling at about 3.34 degrees per minute.
Exercise 21 – Pendulum
\(\theta^{\prime}(t)=0.3\cdot(-\sin 2t)\cdot 2=-0.6\sin 2t\).
At \(t=\frac{\pi}{4}\): \(-0.6\sin\frac{\pi}{2}=-0.6\).
Final answer: \(-0.6\text{ rad/s}\).
Exercise 22 – Nested radicals
Outer: \(\sqrt{u}\) with \(u=x+\sqrt{x}\). Inner derivative: \(1+\frac{1}{2\sqrt{x}}\).
\(f^{\prime}(x)=\frac{1}{2\sqrt{x+\sqrt{x}}}\left(1+\frac{1}{2\sqrt{x}}\right)\).
Final answer: \(\frac{1}{2\sqrt{x+\sqrt{x}}}\left(1+\frac{1}{2\sqrt{x}}\right)\).
Key facts behind this chain rule answers
- 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 answers
Cover each solution, solve the exercise again, then uncover it and compare every line. If your answer differs, find the first step that changed and redo the exercise from there.
Download the chain rule answers as a PDF with the button below and keep it with your notes.
Download the chain rule answers as a PDF with the button below and keep it with your notes.
Download the chain rule answers as a PDF with the button below and keep it with your notes.
Download the chain rule answers as a PDF with the button below and keep it with your notes.
All AP Calculus AB chapters: AP Calculus AB math.
More AP Calculus AB math
Lessons
See all
Worksheets
See all
Tests
See all
Infinite Limits and Asymptotes Test with Point Values- Linearization and L'Hopital's Rule Test with Point Values
- Average Value, Area and Volume Test with Point Values
- Rates of Change in Context and Straight-Line Motion Test with Point Values
- Concavity and the Second Derivative Test with Point Values
- Derivatives of Inverse Functions Test with Point Values
