Calculus · Step-by-step guide

Product Rule Examples

Differentiate products of functions with the product rule and simplify the result.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

When u and v both depend on x, (uv)′=u′v+uv′\displaystyle (uv)'=u'v+uv'. Differentiate one factor at a time while leaving the other unchanged, then add the two products.

(uv)′=u′v+uv′\displaystyle (uv)'=u'v+uv'
01

How it works

Build the method from meaning before memorizing the moves.

These are the decisions that make the procedure reliable. Read them once, then look for each idea in the worked examples below.

1

It is not u′v′

The change in a product has two contributions—one from each changing factor.

2

Label the factors

Writing u, u′, v, and v′ before substituting reduces dropped terms and sign errors.

3

Combine with other rules

Each factor may itself require the chain rule or another derivative formula.

02

See the structure

A picture makes the relationships easier to remember.

tangent line
The derivative at the marked point is the slope of the highlighted tangent line.
03

Worked examples

Follow the reason for each line, then try to reproduce it without looking.

Example 1Polynomial times exponential

Differentiate x2ex\displaystyle x^2e^x.

  1. Let u=x2\displaystyle u=x^2, v=ex\displaystyle v=e^x.
  2. Then u′=2x\displaystyle u'=2x, v′=ex\displaystyle v'=e^x.
  3. Use u′v+uv′\displaystyle u'v+uv'.
Answer2xex+x2ex\displaystyle 2xe^x+x^2e^x
Example 2Polynomial times trig

Differentiate (x+1)sin⁡x\displaystyle (x+1)\sin x.

  1. Differentiate x + 1 to get 1.
  2. Differentiate sine to get cosine.
  3. Add the two product terms.
Answersin⁡x+(x+1)cos⁡x\displaystyle \sin x+(x+1)\cos x
Example 3Product with chain rule

Differentiate x(2x+1)4\displaystyle x(2x+1)^4.

  1. Product rule creates two terms.
  2. The derivative of (2x+1)4\displaystyle (2x+1)^4 is 8(2x+1)3\displaystyle 8(2x+1)^3.
  3. Substitute into the product rule.
Answer(2x+1)4+8x(2x+1)3\displaystyle (2x+1)^4+8x(2x+1)^3
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Multiplying the derivatives only.

  • Using a minus sign between product-rule terms.

  • Forgetting a chain-rule factor inside one product factor.

05

Turn the explanation into a skill

Reading creates recognition. Independent practice creates recall.

Free printable worksheet · Answer key included

Product and Quotient Rule

Practice both rules and learn when each applies.

Preview the worksheet

A strong practice loop: solve one example with the guide open, solve a similar problem without it, explain the method aloud, and return the next day for a short mixed review.

06

Practice problems with worked solutions

Solve each problem on paper, use the hint only if needed, then compare every step.

1

Differentiate x2ex\displaystyle x^2e^x.

Show hint

Use u′v+uv′\displaystyle u'v+uv'.

Show worked solution
  1. u=x2,u′=2x\displaystyle u=x^2,u'=2x; v=ex,v′=ex\displaystyle v=e^x,v'=e^x.
  2. f′=2xex+x2ex\displaystyle f'=2xe^x+x^2e^x.

Answer: ex(x2+2x)\displaystyle e^x(x^2+2x)

2

Differentiate (3x−1)sin⁡x\displaystyle (3x-1)\sin x.

Show hint

Differentiate one factor at a time.

Show worked solution
  1. u′=3\displaystyle u'=3 and v′=cos⁡x\displaystyle v'=\cos x.
  2. f′=3sin⁡x+(3x−1)cos⁡x\displaystyle f'=3\sin x+(3x-1)\cos x.

Answer: 3sin⁡x+(3x−1)cos⁡x\displaystyle 3\sin x+(3x-1)\cos x

3

Differentiate x3ln⁡x\displaystyle x^3\ln x.

Show hint

Use the product rule.

Show worked solution
  1. u′=3x2\displaystyle u'=3x^2 and v′=1/x\displaystyle v'=1/x.
  2. f′=3x2ln⁡x+x3(1/x)\displaystyle f'=3x^2\ln x+x^3(1/x).

Answer: 3x2ln⁡x+x2\displaystyle 3x^2\ln x+x^2

4

Differentiate (x2+1)(x2−1)\displaystyle (x^2+1)(x^2-1).

Show hint

Product rule or simplify first.

Show worked solution
  1. Simplify to x4−1\displaystyle x^4-1.
  2. Differentiate with the power rule.

Answer: 4x3\displaystyle 4x^3

5

Find the tangent slope of f(x)=xcos⁡x\displaystyle f(x)=x\cos x at x = 0.

Show hint

Differentiate, then substitute.

Show worked solution
  1. f′(x)=cos⁡x−xsin⁡x\displaystyle f'(x)=\cos x-x\sin x.
  2. f′(0)=1−0\displaystyle f'(0)=1-0.

Answer: 1\displaystyle 1

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

When u and v both depend on x, (uv)′=u′v+uv′\displaystyle (uv)'=u'v+uv'. Differentiate one factor at a time while leaving the other unchanged, then add the two products.

How do I know whether I understand this topic?

You should be able to name the method, explain why each step is valid, complete a new example without copying, and check whether your answer is reasonable.

What should I do if I keep making the same mistake?

Write the error as a specific checkpoint—for example, “I will identify the hypotenuse before substituting.” Then use that checkpoint on three short problems rather than repeating a full page without feedback.

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