Derivative Practice Problems

30 problems with answers, from the power rule to implicit differentiation and tangent lines. Every answer shows the rule used, so you can see exactly where a mistake came from. Matches AP Calculus AB and BC Units 2 and 3.

30 problems · answer key included · free, no sign-up

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A. Power rule

Find f'(x). Rewrite roots and fractions as powers first.

1
$f (x) = x^{5}$
2
$f (x) = 4 x^{3} - 2 x^{2} + 7 x - 9$
3
$f (x) = x$
4
$f (x) = \frac{3}{x ^{2}}$
5
$f (x) = 6 x^{2/3}$
6
$f (x) = \frac{x ^{3} + 2 x}{x}$

B. Product and quotient rules

Find the derivative.

7
$f (x) = (x^{2} + 1) (3 x - 4)$
8
$f (x) = x^{3} sin x$
9
$f (x) = x^{2} e^{x}$
10
$g (x) = \frac{x}{x + 1}$
11
$h (x) = \frac{x ^{2} - 1}{x ^{2} + 1}$
12
$f (x) = \frac{sin x}{x}$

C. Chain rule

Find the derivative. Identify the inside and outside functions first.

13
$f (x) = (3 x + 2)^{5}$
14
$f (x) = x^{2} + 9$
15
$f (x) = sin (4 x)$
16
$f (x) = e^{x^{2}}$
17
$f (x) = cos^{3} x$
18
$f (x) = ln (x^{2} + 1)$

D. Trig, exponential and logarithmic functions

Find the derivative.

19
$f (x) = tan x + sec x$
20
$f (x) = e^{3 x} - 2^{x}$
21
$f (x) = ln (5 x)$
22
$f (x) = lo g_{2} x$
23
$f (x) = arctan (2 x)$
24
$f (x) = e^{x} cos x$

E. Implicit differentiation

Find dy/dx by differentiating both sides with respect to x.

25
$x^{2} + y^{2} = 25$
26
$x y + y^{2} = 10$
27
$x^{3} + y^{3} = 6 x y$

F. Tangent lines and slopes

Answer each question. Show your derivative.

28
Find the slope of the tangent line to $f (x) = x^{3} - 3 x$ at $x = 2$ .
29
Write the equation of the tangent line to $y = x$ at $x = 4$ .
30
Where does $f (x) = x^{3} - 6 x^{2} + 9 x + 1$ have horizontal tangent lines?

Answer key

Check your work after you finish each section.

A. Power rule

1.
$f^{'} (x) = 5 x^{4}$
Power rule
2.
$f^{'} (x) = 12 x^{2} - 4 x + 7$
Power, constant multiple and sum rules
3.
$f^{'} (x) = \frac{1}{2 x}$
Rewrite as $x^{1/2}$ , then power rule
4.
$f^{'} (x) = - \frac{6}{x ^{3}}$
Rewrite as $3 x^{- 2}$ , then power rule
5.
$f^{'} (x) = 4 x^{- 1/3} = \frac{4}{3 x}$
Power rule with a fractional exponent
6.
$f^{'} (x) = 2 x$
Simplify to $x^{2} + 2$ first, then power rule

B. Product and quotient rules

7.
$f^{'} (x) = 9 x^{2} - 8 x + 3$
Product rule (or expand first)
8.
$f^{'} (x) = 3 x^{2} sin x + x^{3} cos x$
Product rule
9.
$f^{'} (x) = e^{x} (x^{2} + 2 x)$
Product rule
10.
$g^{'} (x) = \frac{1}{( x + 1 ) ^{2}}$
Quotient rule
11.
$h^{'} (x) = \frac{4 x}{( x ^{2} + 1 ) ^{2}}$
Quotient rule
12.
$f^{'} (x) = \frac{x cos x - sin x}{x ^{2}}$
Quotient rule

C. Chain rule

13.
$f^{'} (x) = 15 (3 x + 2)^{4}$
Chain rule: $5 (3 x + 2)^{4} \cdot 3$
14.
$f^{'} (x) = \frac{x}{x ^{2} + 9}$
Chain rule with the power rule
15.
$f^{'} (x) = 4 cos (4 x)$
Chain rule
16.
$f^{'} (x) = 2 x e^{x^{2}}$
Chain rule
17.
$f^{'} (x) = - 3 cos^{2} x sin x$
Chain rule: $3 cos^{2} x \cdot (- sin x)$
18.
$f^{'} (x) = \frac{2 x}{x ^{2} + 1}$
Chain rule with $ln u$

D. Trig, exponential and logarithmic functions

19.
$f^{'} (x) = sec^{2} x + sec x tan x$
Trig derivatives
20.
$f^{'} (x) = 3 e^{3 x} - 2^{x} ln 2$
Use $\frac{d}{d x} a^{x} = a^{x} ln a$
21.
$f^{'} (x) = \frac{1}{x}$
$ln (5 x) = ln 5 + ln x$ , and $ln 5$ is a constant
22.
$f^{'} (x) = \frac{1}{x ln 2}$
Use $\frac{d}{d x} lo g_{a} x = \frac{1}{x l n a}$
23.
$f^{'} (x) = \frac{2}{1 + 4 x ^{2}}$
Inverse tangent with the chain rule
24.
$f^{'} (x) = e^{x} (cos x - sin x)$
Product rule

E. Implicit differentiation

25.
$\frac{d y}{d x} = - \frac{x}{y}$
Implicit differentiation
26.
$\frac{d y}{d x} = - \frac{y}{x + 2 y}$
Implicit differentiation
27.
$\frac{d y}{d x} = \frac{2 y - x ^{2}}{y ^{2} - 2 x}$
Implicit differentiation

F. Tangent lines and slopes

28.
$f^{'} (x) = 3 x^{2} - 3, f^{'} (2) = 9$
Evaluate the derivative at the point
29.
$y - 2 = \frac{1}{4} (x - 4) \Rightarrow y = \frac{1}{4} x + 1$
Point $(4, 2)$ , slope $f^{'} (4) = \frac{1}{4}$
30.
$f^{'} (x) = 3 (x - 1) (x - 3) = 0 \Rightarrow x = 1, x = 3$
Set $f^{'} (x) = 0$ and solve

Which derivative rule do I use?

One term with a power of x: power rule, $\frac{d}{d x} x^{n} = n x^{n - 1}$ . Rewrite roots and fractions first: $x = x^{1/2}$ , $\frac{1}{x ^{3}} = x^{- 3}$ .

Two functions multiplied: product rule, $(f g)^{'} = f^{'} g + f g^{'}$ .

One function divided by another: quotient rule, $(\frac{f}{g})^{'} = \frac{f ^{'} g - f g ^{'}}{g ^{2}}$ . Many students remember it as “low d-high minus high d-low, over low squared.”

A function inside another function: chain rule, $\frac{d}{d x} f (g (x)) = f^{'} (g (x)) \cdot g^{'} (x)$ . Derivative of the outside, times the derivative of the inside.

y mixed in with x: implicit differentiation. Differentiate both sides, attach $\frac{d y}{d x}$ to every y term, then solve for $\frac{d y}{d x}$ .

Need every formula in one place? Use the free AP Calculus formula sheet.

Read the full Derivatives lesson with worked examples →

Frequently asked questions

What are the basic derivative rules?

The power rule (d/dx xⁿ = n xⁿ⁻¹), the constant multiple and sum rules, the product rule (f g)' = f' g + f g', the quotient rule (f/g)' = (f' g − f g')/g², and the chain rule d/dx f(g(x)) = f'(g(x)) · g'(x). Add the derivatives of sin, cos, tan, eˣ and ln x and you can differentiate almost anything on the AP Calculus exam.

How do I know when to use the chain rule?

Use it whenever one function is inside another: a power of an expression like (3x + 2)⁵, a trig function of something other than plain x like sin(4x), or e or ln of an expression. Differentiate the outside function, keep the inside the same, then multiply by the derivative of the inside.

Should I use the quotient rule or rewrite the function?

If the denominator is a single term, rewriting is usually faster: 3/x² is 3x⁻², so the derivative is −6x⁻³. Use the quotient rule when the denominator has more than one term, like x/(x + 1).

What is implicit differentiation?

It is how you find dy/dx when y is not solved for, as in x² + y² = 25. Differentiate both sides with respect to x, treating y as a function of x, so every y term picks up a factor of dy/dx. Then solve for dy/dx.

Are these problems like the AP Calculus exam?

Sections A to E cover the derivative skills in AP Calculus AB and BC Units 2 and 3, and section F matches the tangent-line and slope questions that appear in both multiple choice and free response. Every answer includes the rule used.

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More free worksheets and lessons

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