AP Calculus AB Product Rule, Quotient Rule and Trig Derivatives Test with Point Values (PDF)

AP Calculus AB math test: Product Rule, Quotient Rule and Trig Derivatives – Maths PDF

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This product quotient test for AP Calculus AB covers product rule, quotient rule and trig derivatives (Common Core AP CALC AB 2.8, AP CALC AB 2.9, AP CALC AB 2.10), with a free PDF to print.

This test checks the product rule, the quotient rule and the derivatives of tan x, cot x, sec x and csc x (AP Calculus AB, 2.8, 2.9, 2.10). 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

About this test

Grade: AP Calculus AB
Chapter: Product Rule, Quotient Rule and Trig Derivatives
Standards: AP CALC AB 2.8, AP CALC AB 2.9, AP CALC AB 2.10
Time: 45 minutes
Total: 20 points

Product quotient test: questions

Question 1 – Product rule [2 pts]

Find \(f^{\prime}(x)\) for \(f(x)=(x^{2}+3)e^{x}\).

Question 2 – Quotient rule [3 pts]

Find \(f^{\prime}(x)\) for \(f(x)=\frac{2x-1}{x+4}\).

Question 3 – Trig and product [2 pts]

Differentiate \(g(x)=x\cos x+3\tan x\).

Question 4 – Tangent line [3 pts]

Graph of f(x) = x² ln x.

Find the tangent line to \(f(x)=x^{2}\ln x\) at \(x=e\).

Question 5 – Table of values [2 pts]

Values at x = 1.

Given \(f(1)=2\), \(f^{\prime}(1)=3\), \(g(1)=-1\) and \(g^{\prime}(1)=4\), find \((fg)^{\prime}(1)\) and \(\left(\frac{f}{g}\right)^{\prime}(1)\).

Question 6 – Horizontal tangents [3 pts]

Graph of h(x) = x/(x² + 1).

Find the points where \(h(x)=\frac{x}{x^{2}+1}\) has a horizontal tangent.

Question 7 – Rate of sales [3 pts]

A bakery sells \(n(t)=120-2t\) muffins a day at a price of \(p(t)=2+0.1t\) dollars, where \(t\) is days since a promotion began. (a) Write the revenue \(R(t)=p(t)n(t)\). (1 point) (b) Find \(R^{\prime}(10)\). (2 points)

Question 8 – Secant and cosecant [2 pts]

Differentiate \(k(x)=\sec x-2\csc x\) and find \(k^{\prime}\left(\frac{\pi}{4}\right)\).

Key facts behind this product quotient test

Key facts

Product rule: (uv)′ = u′v + uv′.
Quotient rule: (u/v)′ = (u′v – uv′)/v², and the order in the numerator matters.
d/dx tan x = sec²x and d/dx sec x = sec x tan x.
The derivatives of cot x and csc x are negative: -csc²x and -csc x cot x.

How to use this product quotient 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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