Differentiation Rules Examples in Math

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

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

Concept Recap

A set of standard formulas for finding derivatives of common function types without using the limit definition each time.

Shortcuts so you don't have to use the limit definition every time.

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: Differentiation rules are standard formulas (power, product, quotient) that compute derivatives of common function forms directly.

Common stuck point: The procedure for differentiation rules is the easy part; the trap is thinking the derivative of a product is the product of the derivatives. Asking "Can I name the expression's shape (power, product, or quotient) and apply the matching formula instead of the limit definition?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Can I name the expression's shape (power, product, or quotient) and apply the matching formula instead of the limit definition?

Worked Examples

Example 1

easy

Use the product rule to differentiate

f(x) = x^3 \cdot \sin x

Answer

f'(x) = 3x^2 \sin x + x^3 \cos x

First step

Identify the two factors:

u = x^3

and

v = \sin x

Full solution

2
Find their derivatives: $u' = 3x^2$ and $v' = \cos x$ .
3
Apply the product rule $(uv)' = u'v + uv'$ .
4
Result: $f'(x) = 3x^2 \sin x + x^3 \cos x$ .

The product rule states

(fg)' = f'g + fg'

. Each factor is differentiated once while the other is kept intact, and the two results are added. Never multiply the individual derivatives together.

Example 2

medium

Use the quotient rule to differentiate

f(x) = \dfrac{x^2 + 1}{x - 3}

Example 3

easy

Differentiate

f(x) = 5 \sin x - 2 \cos x

Example 4

medium

Find

f'(x)

when

f(x) = x^2 \ln x

Example 5

hard

Differentiate

f(x) = \dfrac{\sin x}{x^2}

using the quotient rule.

Practice Problems

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

Example 1

easy

Differentiate

f(x) = x^2 \cos x

using the product rule.

Example 2

hard

Differentiate

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

using the quotient rule.

Example 3

easy

Differentiate

f(x) = x^5

using the power rule.

Example 4

easy

Differentiate

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

Example 5

easy

Differentiate

f(x) = 6x

using the constant multiple rule.

Example 6

easy

Differentiate

f(x) = \sin x

Example 7

easy

Differentiate

f(x) = \cos x

Example 8

easy

Differentiate

f(x) = e^x + x^2

Example 9

easy

Differentiate

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

Example 10

easy

Differentiate

f(x) = \ln x

Example 11

medium

Differentiate

f(x) = x^2 \sin x

using the product rule.

Example 12

medium

Differentiate

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

using the quotient rule.

Example 13

medium

Differentiate

f(x) = (3x+1)^5

(power rule plus chain rule).

Example 14

medium

Differentiate

f(x) = x e^x

using the product rule.

Example 15

medium

Differentiate

f(x) = \tan x

(express using

\sec

Example 16

medium

Differentiate

f(x) = (x^2+1)(x-3)

two ways and confirm.

Example 17

medium

Differentiate

f(x) = \frac{e^x}{x}

using the quotient rule.

Example 18

challenge

Differentiate

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

(extended product rule).

Example 19

challenge

Differentiate

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

and simplify.

Example 20

challenge

Find the second derivative of

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

Example 21

medium

Differentiate

f(x) = (2x+1)(x^2 - 3)

using the product rule.

Example 22

medium

Differentiate

f(x) = \frac{3x - 1}{2x + 5}

using the quotient rule.

Example 23

easy

Differentiate

f(x) = 7

Example 24

easy

Differentiate

f(x) = x^{10}

Example 25

easy

Differentiate

f(x) = \sqrt{x}

Example 26

easy

Differentiate

f(x) = x^3 - 4x + 9

Example 27

easy

Differentiate

f(x) = \dfrac{1}{x^3}

Example 28

medium

Differentiate

f(x) = (x^2 + 3)(2x - 1)

using the product rule.

Example 29

medium

Differentiate

f(x) = \dfrac{\ln x}{x}

Example 30

medium

Differentiate

f(x) = (4x - 5)^7

using the chain rule.

Example 31

medium

Differentiate

f(x) = \sin(3x^2)

Example 32

medium

Differentiate

f(x) = e^{5x}

Example 33

medium

Differentiate

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

Example 34

medium

Differentiate

f(x) = \dfrac{x}{x^2 + 4}

Example 35

hard

Differentiate

f(x) = \sin^2 x

Example 36

hard

Differentiate

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

Example 37

hard

Differentiate

f(x) = \tan(x^3)

Example 38

hard

Find

f'(x)

for

f(x) = \sqrt{x^2 + 9}

Example 39

hard

Differentiate

f(x) = x^x

for

x > 0

Example 40

hard

Find the slope of

y = e^{2x}

x = 1

Example 41

hard

Find

f''(x)

for

f(x) = \sin(2x)

Example 42

challenge

Differentiate

f(x) = \ln(\sec x + \tan x)

Example 43

challenge

Use implicit differentiation to find

\dfrac{dy}{dx}

when

x^2 + y^2 = 25

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Related Concepts

Derivative Chain Rule Optimization

Background Knowledge

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

derivative