Calculus · Step-by-step guide

Quotient Rule Examples

Differentiate quotients with the quotient rule, manage signs, and recognize when rewriting is easier.

By Tyler Blovat··9 min read
The short answer

Start with the central idea

For f=uv\displaystyle f=\frac{u}{v}, f′=u′v−uv′v2\displaystyle f'=\frac{u'v-uv'}{v^2}. A memory aid is “low d-high minus high d-low, over low squared.” Keep the numerator order and square the entire denominator.

(uv)′=u′v−uv′v2\displaystyle \left(\frac{u}{v}\right)'=\frac{u'v-uv'}{v^2}
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

Order matters

The numerator begins with denominator times derivative of numerator, then subtracts numerator times derivative of denominator.

2

Square the denominator

The original entire denominator—not just one term—is squared.

3

Rewrite when helpful

A simple power of x in the denominator may be easier to rewrite with a negative exponent.

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 1Rational function

Differentiate x2+1x\displaystyle \frac{x^2+1}{x}.

  1. u′ = 2x and v′ = 1.
  2. Numerator: (2x)(x)−(x2+1)(1)\displaystyle (2x)(x)-(x^2+1)(1).
  3. Denominator: x²; simplify.
Answer1−1x2\displaystyle 1-\frac{1}{x^2}
Example 2Trig quotient

Differentiate sin⁡xx+1\displaystyle \frac{\sin x}{x+1}.

  1. u′ = cos x; v′ = 1.
  2. Keep the required subtraction order.
  3. Square x + 1 in parentheses.
Answer(x+1)cos⁡x−sin⁡x(x+1)2\displaystyle \frac{(x+1)\cos x-\sin x}{(x+1)^2}
Example 3Rewrite first

Differentiate 5x3\displaystyle \frac{5}{x^3}.

  1. Rewrite as 5x−3\displaystyle 5x^{-3}.
  2. Apply the power rule.
  3. Return to positive exponents.
Answer−15x4\displaystyle -\frac{15}{x^4}
04

Common mistakes—and how to avoid them

Accuracy improves fastest when you know what to check.

  • Reversing the two numerator products.

  • Forgetting to square the denominator.

  • Canceling terms across addition or subtraction.

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 rule selection, setup, and simplification.

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 x2+1x\displaystyle \frac{x^2+1}{x}.

Show hint

Simplify first or use the quotient rule.

Show worked solution
  1. Rewrite as x+x−1\displaystyle x+x^{-1}.
  2. Differentiate term by term.

Answer: 1−1x2\displaystyle 1-\frac{1}{x^2}

2

Differentiate 2x+3x−1\displaystyle \frac{2x+3}{x-1}.

Show hint

Use low d-high minus high d-low.

Show worked solution
  1. Numerator: 2(x−1)−(2x+3)(1)\displaystyle 2(x-1)-(2x+3)(1).
  2. Simplify to 2x−2−2x−3=−5\displaystyle 2x-2-2x-3=-5.

Answer: −5(x−1)2\displaystyle -\frac{5}{(x-1)^2}

3

Differentiate sin⁡xx2\displaystyle \frac{\sin x}{x^2}.

Show hint

Square the full denominator.

Show worked solution
  1. f′=x2cos⁡x−2xsin⁡xx4\displaystyle f'=\frac{x^2\cos x-2x\sin x}{x^4}.
  2. Cancel one x where possible.

Answer: xcos⁡x−2sin⁡xx3\displaystyle \frac{x\cos x-2\sin x}{x^3}

4

Differentiate exx+1\displaystyle \frac{e^x}{x+1}.

Show hint

Keep the subtraction order.

Show worked solution
  1. f′=(x+1)ex−ex(x+1)2\displaystyle f'=\frac{(x+1)e^x-e^x}{(x+1)^2}.
  2. Factor and simplify the numerator.

Answer: xex(x+1)2\displaystyle \frac{xe^x}{(x+1)^2}

5

Find the slope of f(x)=xx+2\displaystyle f(x)=\frac{x}{x+2} at x = 1.

Show hint

Find f′ first.

Show worked solution
  1. f′=(x+2)−x(x+2)2=2(x+2)2\displaystyle f'=\frac{(x+2)-x}{(x+2)^2}=\frac{2}{(x+2)^2}.
  2. f′(1)=29\displaystyle f'(1)=\frac{2}{9}.

Answer: 29\displaystyle \frac{2}{9}

07

Frequently asked questions

Quick answers before you move on.

What should I remember first?

For f=uv\displaystyle f=\frac{u}{v}, f′=u′v−uv′v2\displaystyle f'=\frac{u'v-uv'}{v^2}. A memory aid is “low d-high minus high d-low, over low squared.” Keep the numerator order and square the entire denominator.

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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