Quotient Rule Calculator

Published on: April 27, 2025
Final Answer: Free • Full Steps: Plus

This Quotient Rule Calculator helps you differentiate quotients such as (x^2 + 1)/(x - 1) or sin(x)/x. The quotient rule combines the derivatives of the numerator and denominator into a single fraction over the denominator squared.

Step-by-step method

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

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

Example 1:

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

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 - 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 - 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 - 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 - 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 - 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:

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

Example 2:

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

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 - 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 - 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 - Differentiate the denominator.

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

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

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 - 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:

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