Quotient Rule

If the two functions f(x) and g(x) are differentiable (i.e., the derivative exist) then the quotient is differentiable. The quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions.

The quotient rule is a method for finding the derivative of a function that is the ratio of two differentiable functions. The formula is:

$\frac{d}{dx}[u(x)/v(x)]=\,\,\frac{[u\prime(x)*v(x)\,\,-\,\,u(x)*v\prime(x)]}{[v(x)]^2}$

Where,

u(x) is the first function which is a differentiable function,
u‘(x) is the derivative of function u(x),
v(x) is the second function which is a differentiable function, and
v‘(x) is the derivative of the function v(x).

Many calculus students remember this rule with the mnemonic “Low d-High minus High d-Low, over the square of what’s below”:

Low: The bottom function, v(x)
d-High: The derivative of the top function, u‘(x)
High: The top function, u(x)
d-Low: The derivative of the bottom function, v‘(x)
Low squared: The bottom function squared, [v(x)]²

Quotient Rule Proof

Various proofs of the quotient rule can be found on the BJYU’s and CUEMATH websites, and in the YouTube video below.

Example

Find the derivative of

$h\left( x \right) \,\,=\,\,\frac{x^2}{x\,\,+\,\,1}$

Identify the Pieces

f(x) = x² ⇒ f‘(x) = 2x
g(x) = x + 1 ⇒ g‘(x) = 1

Apply the Formula

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

Simplify

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

References

Calculus. “2.9 The Quotient Rule.” Accessed June 13, 2026. https://calculus.flippedmath.com/29-the-quotient-rule.html.

CUEMATH. “Quotient Rule – Formula, Proof, Definition, Examples.” Accessed June 13, 2026. https://www.cuemath.com/calculus/quotient-rule/.

Dawkins, Paul. “Calculus I.” Product and Quotient Rule. Paul’s Online Notes, November 16, 2022. https://tutorial.math.lamar.edu/classes/calci/productquotientrule.aspx.

GeeksforGeeks. “How to Use Quotient Rule?” GeeksforGeeks, May 21, 2022. https://www.geeksforgeeks.org/maths/how-to-use-quotient-rule/.

GeeksforGeeks. “Quotient Rule.” GeeksforGeeks, May 17, 2021. https://www.geeksforgeeks.org/maths/quotient-rule/.

Guichard, David. “3.4: The Quotient Rule.” LibreTexts, November 7, 2013. https://math.libretexts.org/Bookshelves/Calculus/Calculus_(Guichard)/03%3A_Rules_for_Finding_Derivatives/3.04%3A_The_Quotient_Rule.

“Product and Quotient Rules.” Accessed June 13, 2026. https://web.ma.utexas.edu/users/m408n/CurrentWeb/LM3-2-4.php.

“Quotient Rule.” BYJU’s, January 12, 2018. https://byjus.com/maths/quotient-rule/.

“The Quotient Rule for Derivatives.” SubjectCoach. Accessed June 13, 2026. https://www.subjectcoach.com/tutorials/math/topic/calculus/chapter/the-quotient-rule-for-derivatives.

Videos

Quotient Rule for Derivatives – Calculus 1 Example Problems

Quotient Rule For Derivatives

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Proof of Quotient Rule

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