Practice Derivative in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

The instantaneous rate of change of a function at a single point, defined as the limit of the slope of secant lines.

How fast is the output changing right now? The slope of the curve at each point.

Showing a random 20 of 50 problems.

Example 1

easy

Find

\tfrac{d}{dx}[\tan x]

Example 2

medium

Find the derivative of

f(x) = \frac{x}{x+1}

using the quotient rule.

Example 3

challenge

Find the derivative of

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

Example 4

medium

Use the limit definition to find

f'(x)

for

f(x) = x^2

Example 5

medium

Find the slope of

f(x) = x^2 - 4x

x = 3

Example 6

hard

Find the derivative of

f(x) = x^3 \sin(x) - \frac{e^x}{x^2}

and evaluate

f'(\pi)

Example 7

medium

Find

f'(x)

for

f(x) = \sqrt{x} \cdot \ln x

Example 8

easy

Find the derivative of

f(x) = 3x^2 + 5x - 2

Example 9

medium

Find the derivative of

f(x) = \cos x

and evaluate at

x = 0

Example 10

easy

Find the derivative of

f(x) = 4x^2

Example 11

easy

Find

f'(x)

for

f(x) = \ln x

Example 12

challenge

Use the limit definition to show

\frac{d}{dx}\frac{1}{x} = -\frac{1}{x^2}

Example 13

easy

What is the slope of

f(x) = x^2

x = -3

Example 14

medium

Find the derivative of

f(x) = \sin x \cos x

Example 15

medium

Find the equation of the tangent line to

y = x^3

x = 2

Example 16

easy

Find

\tfrac{d}{dx}[\cos x]

Example 17

medium

Find

f'(x)

for

f(x) = x e^x

Example 18

hard

Use implicit differentiation to find

\tfrac{dy}{dx}

for

x^2 + y^2 = 25

Example 19

medium

Find the derivative of

f(x) = x^2 e^x

using the product rule.

Example 20

easy

Find the derivative of

f(x) = 5x

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Related Concepts

Limit Slope Differentiation Rules Chain Rule Integral