AP Calculus AB Power, Sum and Basic Function Derivative Rules Test with Point Values (PDF)

AP Calculus AB math test: Power, Sum and Basic Function Derivative Rules – Maths PDF

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This derivative rules test for AP Calculus AB covers power, sum and basic function derivative rules (Common Core AP CALC AB 2.5, AP CALC AB 2.6, AP CALC AB 2.7), with a free PDF to print.

This test checks the power rule, sum and constant multiple rules, derivatives of sin x, cos x, e^x and ln x, and tangent and normal lines (AP Calculus AB, 2.5, 2.6, 2.7). 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: Power, Sum and Basic Function Derivative Rules
Standards: AP CALC AB 2.5, AP CALC AB 2.6, AP CALC AB 2.7
Time: 45 minutes
Total: 20 points

Derivative rules test: questions

Question 1 – Polynomial derivative [2 pts]

Find \(f^{\prime}(x)\) for \(f(x)=5 x^{4} – 3 x^{3} + 2 x – 8\).

Question 2 – Roots and reciprocals [2 pts]

Find \(f^{\prime}(x)\) for \(f(x)=4\sqrt{x}-\frac{3}{x}+x^{5/2}\).

Question 3 – Trig, exponential and log [2 pts]

Find \(\frac{d}{dx}\left[2\cos x+3e^{x}-\ln x\right]\).

Question 4 – Tangent line [3 pts]

Graph of f(x) = x³ − 3x² + 2.

Find the equation of the tangent line to \(f(x)=x^{3}-3x^{2}+2\) at \(x=1\).

Question 5 – Horizontal tangents [3 pts]

Graph of g(x) = x³ − 6x² + 9x + 1.

Find the points where the graph of \(g(x)=x^{3}-6x^{2}+9x+1\) has a horizontal tangent line.

Question 6 – Normal line [2 pts]

Graph of f(x) = x² + 1 and the normal line at x = 2.

Find the equation of the normal line to \(f(x)=x^{2}+1\) at \(x=2\).

Question 7 – Velocity in context [3 pts]

A ball rolls so that its position in feet is \(s(t)=t^{3}-9t^{2}+15t\) after \(t\) seconds. (a) Find the velocity \(v(t)\). (1 point) (b) Find the velocity at \(t=2\). (1 point) (c) When is the ball at rest? (1 point)

Question 8 – Unknown constants [3 pts]

The function \(f(x)=x^{2}+bx+c\) has the tangent line \(y=3x+1\) at \(x=1\). Find \(b\) and \(c\).

Key facts behind this derivative rules test

Key facts

Power rule: d/dx of x^n equals n x^(n-1); the derivative of a constant is 0.
Rewrite roots and reciprocals as powers before differentiating.
The derivatives of sin x, cos x, e^x and ln x are cos x, -sin x, e^x and 1/x.
A normal line has slope equal to the negative reciprocal of the tangent slope.

How to use this derivative rules 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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