Quotient Rule Notes

Published on: September 29, 2026

This Quotient Rule Calculator differentiates one function divided by another, and shows the working line by line. The numerator of the answer subtracts in a fixed order, so which function is on top matters and swapping them flips the sign. The denominator is squared. Following the same pattern every time keeps the order of subtraction and the squared denominator from being muddled.

Step-by-step method

  1. Set up the problem.
  2. Write the quotient rule formula.
  3. Differentiate the numerator.
  4. Differentiate the denominator.
  5. Substitute into the quotient rule formula.
  6. Simplify the result.

Formula:

\(\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right) = \frac{f'\left(x\right)g\left(x\right) - f\left(x\right)g'\left(x\right)}{\left[g\left(x\right)\right]^{2}}\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the quotient rule formula.
  3. Differentiate the numerator.
  4. Differentiate the denominator.
  5. Substitute into the quotient rule formula.
  6. Simplify the result.

Step 1

Step 1 - Set up the numerator f(x) and the denominator g(x).

In this problem: The numerator is \(x^{2} + 1\) and the denominator is \(x - 1\).

\(\begin{gathered} f\left(x\right) = x^{2} + 1 \\ g\left(x\right) = x - 1 \end{gathered}\)

Step 2

Step 2 - Write the quotient rule formula.

In this problem: The derivative of a quotient is: bottom times derivative of top, minus top times derivative of bottom, all over the bottom squared.

\(\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right) = \frac{f'\left(x\right)g\left(x\right) - f\left(x\right)g'\left(x\right)}{\left[g\left(x\right)\right]^{2}}\)

Step 3

Step 3 - Differentiate the numerator.

In this problem: Differentiate \(x^{2} + 1\) to get \(2 x\).

\(f'\left(x\right) = \frac{d}{dx}\left(x^{2} + 1\right) = 2 x\)

Step 4

Step 4 - Differentiate the denominator.

In this problem: Differentiate \(x - 1\) to get \(1\).

\(g'\left(x\right) = \frac{d}{dx}\left(x - 1\right) = 1\)

Step 5

Step 5 - Substitute into the quotient rule formula.

In this problem: Substitute the functions and their derivatives into the formula.

\(y'\left(x\right) = \frac{\left(2 x\right)\left(x - 1\right) - \left(x^{2} + 1\right)\left(1\right)}{\left(x - 1\right)^{2}}\)

Step 6

Step 6 - Simplify the result.

In this problem: Expand the numerator and simplify the fraction.

\(y'\left(x\right) = \frac{x^{2} - 2 x - 1}{\left(x - 1\right)^{2}} = \frac{x^{2} - 2 x - 1}{x^{2} - 2 x + 1}\)

Final Answer

\(f'\left(x\right) = \frac{x^{2} - 2 x - 1}{x^{2} - 2 x + 1}\)
See Example 2 Hide Example 2

Problem

\(f\left(x\right) = \frac{\sin{\left(x \right)}}{x}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the quotient rule formula.
  3. Differentiate the numerator.
  4. Differentiate the denominator.
  5. Substitute into the quotient rule formula.
  6. Simplify the result.

Step 1

Step 1 - Set up the numerator f(x) and the denominator g(x).

In this problem: The numerator is \(\sin{\left(x \right)}\) and the denominator is \(x\).

\(\begin{gathered} f\left(x\right) = \sin{\left(x \right)} \\ g\left(x\right) = x \end{gathered}\)

Step 2

Step 2 - Write the quotient rule formula.

In this problem: The derivative of a quotient is: bottom times derivative of top, minus top times derivative of bottom, all over the bottom squared.

\(\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right) = \frac{f'\left(x\right)g\left(x\right) - f\left(x\right)g'\left(x\right)}{\left[g\left(x\right)\right]^{2}}\)

Step 3

Step 3 - Differentiate the numerator.

In this problem: Differentiate \(\sin{\left(x \right)}\) to get \(\cos{\left(x \right)}\).

\(f'\left(x\right) = \frac{d}{dx}\left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}\)

Step 4

Step 4 - Differentiate the denominator.

In this problem: Differentiate \(x\) to get \(1\).

\(g'\left(x\right) = \frac{d}{dx}\left(x\right) = 1\)

Step 5

Step 5 - Substitute into the quotient rule formula.

In this problem: Substitute the functions and their derivatives into the formula.

\(y'\left(x\right) = \frac{\left(\cos{\left(x \right)}\right)\left(x\right) - \left(\sin{\left(x \right)}\right)\left(1\right)}{\left(x\right)^{2}}\)

Step 6

Step 6 - Simplify the result.

In this problem: Expand the numerator and simplify the fraction.

\(y'\left(x\right) = \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{\left(x\right)^{2}} = \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}}\)

Final Answer

\(f'\left(x\right) = \frac{x \cos{\left(x \right)} - \sin{\left(x \right)}}{x^{2}}\)
See Example 3 Hide Example 3

Problem

\(f\left(x\right) = \frac{2 x + 1}{x + 3}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the quotient rule formula.
  3. Differentiate the numerator.
  4. Differentiate the denominator.
  5. Substitute into the quotient rule formula.
  6. Simplify the result.

Step 1

Step 1 - Set up the numerator f(x) and the denominator g(x).

In this problem: The numerator is \(2 x + 1\) and the denominator is \(x + 3\).

\(\begin{gathered} f\left(x\right) = 2 x + 1 \\ g\left(x\right) = x + 3 \end{gathered}\)

Step 2

Step 2 - Write the quotient rule formula.

In this problem: The derivative of a quotient is: bottom times derivative of top, minus top times derivative of bottom, all over the bottom squared.

\(\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right) = \frac{f'\left(x\right)g\left(x\right) - f\left(x\right)g'\left(x\right)}{\left[g\left(x\right)\right]^{2}}\)

Step 3

Step 3 - Differentiate the numerator.

In this problem: Differentiate \(2 x + 1\) to get \(2\).

\(f'\left(x\right) = \frac{d}{dx}\left(2 x + 1\right) = 2\)

Step 4

Step 4 - Differentiate the denominator.

In this problem: Differentiate \(x + 3\) to get \(1\).

\(g'\left(x\right) = \frac{d}{dx}\left(x + 3\right) = 1\)

Step 5

Step 5 - Substitute into the quotient rule formula.

In this problem: Substitute the functions and their derivatives into the formula.

\(y'\left(x\right) = \frac{\left(2\right)\left(x + 3\right) - \left(2 x + 1\right)\left(1\right)}{\left(x + 3\right)^{2}}\)

Step 6

Step 6 - Simplify the result.

In this problem: Expand the numerator and simplify the fraction.

\(y'\left(x\right) = \frac{5}{\left(x + 3\right)^{2}} = \frac{5}{x^{2} + 6 x + 9}\)

Final Answer

\(f'\left(x\right) = \frac{5}{x^{2} + 6 x + 9}\)
See Example 4 Hide Example 4

Problem

\(f\left(x\right) = \frac{\cos{\left(x \right)}}{x^{2}}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Write the quotient rule formula.
  3. Differentiate the numerator.
  4. Differentiate the denominator.
  5. Substitute into the quotient rule formula.
  6. Simplify the result.

Step 1

Step 1 - Set up the numerator f(x) and the denominator g(x).

In this problem: The numerator is \(\cos{\left(x \right)}\) and the denominator is \(x^{2}\).

\(\begin{gathered} f\left(x\right) = \cos{\left(x \right)} \\ g\left(x\right) = x^{2} \end{gathered}\)

Step 2

Step 2 - Write the quotient rule formula.

In this problem: The derivative of a quotient is: bottom times derivative of top, minus top times derivative of bottom, all over the bottom squared.

\(\frac{d}{dx}\left(\frac{f\left(x\right)}{g\left(x\right)}\right) = \frac{f'\left(x\right)g\left(x\right) - f\left(x\right)g'\left(x\right)}{\left[g\left(x\right)\right]^{2}}\)

Step 3

Step 3 - Differentiate the numerator.

In this problem: Differentiate \(\cos{\left(x \right)}\) to get \(- \sin{\left(x \right)}\).

\(f'\left(x\right) = \frac{d}{dx}\left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}\)

Step 4

Step 4 - Differentiate the denominator.

In this problem: Differentiate \(x^{2}\) to get \(2 x\).

\(g'\left(x\right) = \frac{d}{dx}\left(x^{2}\right) = 2 x\)

Step 5

Step 5 - Substitute into the quotient rule formula.

In this problem: Substitute the functions and their derivatives into the formula.

\(y'\left(x\right) = \frac{\left(- \sin{\left(x \right)}\right)\left(x^{2}\right) - \left(\cos{\left(x \right)}\right)\left(2 x\right)}{\left(x^{2}\right)^{2}}\)

Step 6

Step 6 - Simplify the result.

In this problem: Expand the numerator and simplify the fraction.

\(y'\left(x\right) = \frac{- x^{2} \sin{\left(x \right)} - 2 x \cos{\left(x \right)}}{\left(x^{2}\right)^{2}} = - \frac{x \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{x^{3}}\)

Final Answer

\(f'\left(x\right) = - \frac{x \sin{\left(x \right)} + 2 \cos{\left(x \right)}}{x^{3}}\)