Derivative Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Derivative.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept 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.

Read the full concept explanation →

How to Use These Examples

Read the first worked example with the solution open so the structure is clear.
Try the practice problems before revealing each solution.
Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The derivative is the instantaneous rate of change at a single point, found as the limit of secant slopes as the two points merge.

Common stuck point: The procedure for derivative is the easy part; the trap is forgetting the limit and just computing $\frac{f(x+h)-f(x)}{h}$ with a fixed $h$ . Asking "Am I asked for the rate of change at a single instant, found by letting the gap $h$ shrink to zero?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I asked for the rate of change at a single instant, found by letting the gap $h$ shrink to zero?

Common Mistakes to Watch For

Before you work through the examples, skim the mistake guide so you know which shortcuts and sign errors to avoid.

Chain Rule Mistakes Product Rule Mistakes

Worked Examples

Example 1

easy

Find the derivative of

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

Answer

f'(x) = 6x + 5

First step

Apply the power rule to each term:

\frac{d}{dx}[x^n] = nx^{n-1}

Full solution

2
For $3x^2$ : $3 \cdot 2x^{2-1} = 6x$ .
3
For $5x$ : $5 \cdot 1x^{1-1} = 5$ .
4
For $-2$ : the derivative of a constant is $0$ .
5
Combine: $f'(x) = 6x + 5$ .

The power rule is the most fundamental differentiation rule. Apply it term by term and remember that the derivative of a constant is zero.

Example 2

medium

Find the derivative of

f(x) = x^3 - 4x^2 + 7x

and evaluate

f'(2)

Example 3

hard

Use the limit definition to find the derivative of

f(x) = x^2

Example 4

medium

Find

f'(x)

for

f(x) = x^2 \cdot \sin x

Example 5

medium

Find the equation of the tangent line to

y = x^3

x = 2

Example 6

medium

Find the critical points of

f(x) = x^3 - 3x

Example 7

medium

A particle has position

s(t) = t^3 - 6t^2 + 9t

. Find its velocity at

t = 2

Example 8

hard

Use the limit definition to find

f'(x)

for

f(x) = 1/x

Example 9

hard

Find the equation of the tangent line to

y = \ln x

at the point where

y = 0

Example 10

hard

Find the absolute maximum of

f(x) = -x^2 + 4x + 1

[0, 5]

Example 11

challenge

Find the equation of the normal line to

y = x^3 - 2x

(1, -1)

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy

Find the derivative of

f(x) = 7x^4 - 2x + 9

Example 2

hard

Find the derivative of

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

and evaluate

f'(\pi)

Example 3

easy

Find the derivative of

f(x) = x^3

Example 4

easy

Find the derivative of

f(x) = 5x

Example 5

easy

Find the derivative of

f(x) = 7

Example 6

easy

Find the derivative of

f(x) = x^2 + 3x

Example 7

easy

Find the derivative of

f(x) = 4x^2

Example 8

easy

Find the derivative of

f(x) = x^{-2}

Example 9

easy

Find the derivative of

f(x) = \sqrt{x}

Example 10

easy

Find the derivative of

f(x) = e^x

Example 11

medium

Find the derivative of

f(x) = x^2 e^x

using the product rule.

Example 12

medium

Find the derivative of

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

using the quotient rule.

Example 13

medium

Compute the derivative of

f(x) = 3x^4 - 2x^2 + 7

Example 14

medium

Use the limit definition to find

f'(x)

for

f(x) = x^2

Example 15

medium

Find the slope of

f(x) = x^2 - 4x

x = 3

Example 16

medium

Find the derivative of

f(x) = \sin x \cos x

Example 17

medium

Find the derivative of

f(x) = \ln x

and evaluate at

x = 2

Example 18

challenge

Find the derivative of

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

Example 19

challenge

f(x) = ax^2 + bx

has

f'(1) = 5

and

f'(0) = 1

, find

a

and

b

Example 20

challenge

Use the limit definition to show

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

Example 21

medium

Find the derivative of

f(x) = \cos x

and evaluate at

x = 0

Example 22

medium

Find the derivative of

f(x) = x^3 - 6x^2 + 9x

and its critical points.

Example 23

easy

Find

f'(x)

for

f(x) = 6x^5

Example 24

easy

Find

f'(x)

for

f(x) = e^x + 2x

Example 25

easy

What is the slope of

f(x) = x^2

x = -3

Example 26

easy

Find

f'(x)

for

f(x) = 3x^{2/3}

Example 27

medium

Find

f'(x)

for

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

Example 28

medium

Find

f'(x)

for

f(x) = x e^x

Example 29

medium

Find

f'(x)

for

f(x) = \ln(x^2 + 1)

Example 30

medium

Find

f'(x)

for

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

Example 31

hard

Find

f'(x)

for

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

Example 32

hard

Use implicit differentiation to find

\tfrac{dy}{dx}

for

x^2 + y^2 = 25

Example 33

hard

Find

f'(0)

for

f(x) = e^{2x} \cos x

← Back to Derivative Formula Explained Common Mistakes Practice Mode

Related Concepts

Limit Slope Differentiation Rules Chain Rule Integral

Background Knowledge

These ideas may be useful before you work through the harder examples.

limitslope