Definition of the Derivative and Basic Rules: AP Calculus AB Math Worksheet – PDF Download

AP Calculus Derivative Rules Worksheet: problems 1–5

Problem 1: Average rate of change

The graph of \(f(x)=x^{2}+3x\) is shown. Find the average rate of change of \(f\) on \([1,4]\), and write what it represents on the graph.

Problem 2: Derivative at a point by definition

Use the limit definition to find each derivative.

1. \(f^{\prime}(3)\) for \(f(x)=x^{2}\)
2. \(f^{\prime}(2)\) for \(f(x)=x^{2}+4x\)

Problem 3: Limit definition for a function

Let \(f(x)=2x^{2}-5x\). Use the limit definition to find \(f^{\prime}(x)\), then evaluate \(f^{\prime}(1)\).

Problem 4: Definition with a reciprocal

Apply the definition of the derivative as a limit to differentiate \(g(x)=\frac{2}{x-1}\).

Problem 5: Power rule

Differentiate each function.

1. \(f(x)=x^{8}\)
2. \(f(x)=6x^{5}\)
3. \(f(x)=\frac{3}{x^{2}}\)
4. \(f(x)=\sqrt[3]{x}\)

AP Calculus Derivative Rules Worksheet: problems 6–10

Problem 6: Sum and constant multiple rules

Let \(f(x)=4x^{3}-7x^{2}+5x-9\). Find \(f^{\prime}(x)\), evaluate \(f^{\prime}(2)\), and find every \(x\) where \(f^{\prime}(x)=5\).

Problem 7: Derivatives of sin, cos, e^x and ln x

Find the derivative.

1. \(f(x)=3\sin x-2\cos x\)
2. \(f(x)=5e^{x}+2\ln x\)
3. \(g(x)=e^{x}+\sin x\); then find \(g^{\prime}(0)\)
4. \(h(x)=4\ln x-x^{3}\); then find \(h^{\prime}(1)\)

Problem 8: Tangent line

The graph of \(f(x)=x^{3}-2x\) is shown. Write an equation of the tangent line at \(x=2\).

Problem 9: Horizontal tangent lines

At which points does the graph of \(f(x)=2x^{3}-3x^{2}-12x+1\) have a horizontal tangent line? Give both coordinates of each point.

Problem 10: Product rule

Find the derivative. Do not simplify part (a) before checking it.

1. \(f(x)=(x^{2}+1)(x^{3}-4x)\)
2. \(f(x)=x^{2}\sin x\)
3. \(f(x)=e^{x}\ln x\)

AP Calculus Derivative Rules Worksheet: problems 11–15

Problem 11: Quotient rule

Find the derivative.

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

Problem 12: Derivatives of tan, sec, csc and cot

Find the derivative.

1. \(f(x)=\tan x\); then evaluate \(f^{\prime}(\frac{\pi}{4})\)
2. \(f(x)=3\,\text{sec}\,x\)
3. \(f(x)=2\,\text{cot}\,x+\text{csc}\,x\)

Problem 13: Estimate derivatives from a table

Selected values of a differentiable function \(f\) are shown.

x	1	1.5	2	2.5	3
f(x)	2.0	3.1	5.0	7.7	11.0

1. Find the average rate of change of \(f\) on \([1,2]\).
2. Estimate \(f^{\prime}(2)\) using the values at \(x=1.5\) and \(x=2.5\).
3. Use your estimate to write an equation of the tangent line at \(x=2\).

Problem 14: Is it differentiable?

Two graphs are shown: \(f(x)=|x-2|\) and \(p(x)=x^{2}\) for \(x\le 1\), \(p(x)=2x-1\) for \(x > 1\).

1. Is \(f\) differentiable at \(x=2\)? Use one-sided slopes.
2. Show that \(p\) is differentiable at \(x=1\) and find \(p^{\prime}(1)\).

Problem 15: Height of a ball

A ball is thrown upward and its height in feet after \(t\) seconds is \(h(t)=-16t^{2}+48t+5\).

1. Find the average velocity on \([1,2]\).
2. Find the velocity at \(t=1\).
3. Find the time and height at the top of the flight.

AP Calculus Derivative Rules Worksheet: problems 16–20

Problem 16: Marginal cost of bicycles

A shop pays \(C(x)=0.02x^{2}+3x+500\) dollars to build \(x\) bicycle helmets. Find \(C^{\prime}(100)\) and interpret it. Then compare it with \(C(101)-C(100)\).

Problem 17: Learning curve

After \(t\) hours of practice (\(t\ge 1\)), a typing student scores \(S(t)=40+15\ln t\) words per minute. Find \(S^{\prime}(t)\), then \(S^{\prime}(5)\) and \(S^{\prime}(10)\), and explain what the change says about practice.

Problem 18: Revenue growth

A store sells \(q(t)=300-4t\) scarves per day at a price of \(p(t)=20+0.5t\) dollars, where \(t\) is days since a promotion began. Revenue is \(R(t)=p(t)\,q(t)\). Find \(R^{\prime}(10)\) and interpret it.

Problem 19: Concentration of a medicine

The concentration of a medicine in the blood is \(C(t)=\frac{t}{t^{2}+4}\) mg/L, \(t\) hours after a dose.

1. Find \(C^{\prime}(t)\).
2. Find \(C^{\prime}(1)\) and interpret it.
3. At what time is the concentration greatest, and what is the maximum?

Problem 20: Derivatives from a table of values

Values of two differentiable functions are shown.

x	f(x)	f'(x)	g(x)	g'(x)
2	3	-1	5	4

Find the derivative at \(x=2\) of each function.

1. \(h(x)=f(x)\,g(x)\)
2. \(k(x)=\frac{f(x)}{g(x)}\)
3. \(m(x)=2f(x)-3g(x)\)

How to use this ap calculus derivative rules worksheet

Print this ap calculus derivative rules worksheet as a PDF, solve every problem on paper, then compare with the answer key.

See every chapter on the AP Calculus AB math page.