AP Calculus AB The Chain Rule Test with Point Values (PDF)
This chain rule test for AP Calculus AB covers the chain rule (Common Core AP CALC AB 3.1), with a free PDF to print.
This test checks the chain rule for composite functions, including powers, trig, exponential and logarithmic layers (AP Calculus AB, 3.1). It has 8 questions worth 20 points in total. Allow 45 minutes, work without notes, and show all work, since justification earns points on the AP exam.
Time and points
- Grade: AP Calculus AB
- Chapter: The Chain Rule
- Standards: AP CALC AB 3.1
- Time: 45 minutes
- Total: 20 points
Chain rule test: questions
Question 1 – Power chain [2 pts]
Differentiate \(f(x)=(4x-7)^{6}\).
Question 2 – Trig chain [2 pts]
Differentiate \(g(x)=\sin(x^{3})+\cos(5x)\).
Question 3 – Exponential and log chain [2 pts]
Differentiate \(h(x)=e^{x^{2}-3x}+\ln(4x+1)\).
Question 4 – Three layers [3 pts]
Differentiate \(k(x)=\sin^{3}(x^{2}+1)\).
Question 5 – Tangent line [3 pts]
Find the tangent line to \(y=\sqrt{2x^{2}+1}\) at \(x=1\).
Question 6 – Composite from a table [3 pts]
Use the table. (a) Find \((f\circ g)^{\prime}(1)\) where \((f\circ g)(x)=f(g(x))\). (2 points) (b) Find the derivative of \(p(x)=[f(x)]^{2}\) at \(x=4\). (1 point)
Question 7 – Product and chain [2 pts]
Differentiate \(p(x)=x\,e^{3x}\).
Question 8 – Tide height [3 pts]
The height of the tide in feet at a harbor is \(h(t)=12+4\sin\left(\frac{\pi t}{6}\right)\) after \(t\) hours. (a) Find \(h^{\prime}(t)\). (1 point) (b) Find \(h^{\prime}(0)\) and \(h^{\prime}(3)\). (1 point) (c) Interpret \(h^{\prime}(3)\). (1 point)
Key facts behind this chain rule test
- 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 test
Time yourself: the test is designed for an average student in 45 minutes. Answer the questions you are sure of first, then come back to the longer ones and check your units.
All AP Calculus AB chapters: AP Calculus AB math.
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