Calculus – Quotient Rule

The Quotient Rule

The quotient rule is a fundamental rule in calculus used to find the derivative of a function that is expressed as the ratio of two other differentiable functions.

The quotient rule says that the derivative of the quotient is the denominator times the derivative of the numerator minus the numerator times the derivative of the denominator, all divided by the square of the denominator.

The following diagrams show the Quotient Rule used to find the derivative of the division of two functions. Scroll down the page for more examples and solutions on how to use the Quotient Rule.

 

If both \(f(x)\) and \(g(x)\) are differentiable, and \(g(x) \neq 0\), then

 

Mnemonic:
A common mnemonic to remember the quotient rule is:
“Low d-High minus High d-Low, over Low squared."
Where:
“Low” represents the denominator \(g(x)\).
“d-High” represents the derivative of the numerator \(f’(x)\).
“High” represents the numerator \(f(x)\).
“d-Low” represents the derivative of the denominator \(g’(x)\).
“Low squared” represents the denominator squared \([g(x)]^2\).

Important Note: The order of the terms in the numerator of the quotient rule is crucial due to the subtraction. Always remember “Low d-High minus High d-Low”.

Example:
Given that , find f ‘(x)

Solution:

Example:
Given that , find f ‘(x)

Solution:

Why the quotient rule is the same thing as the product rule?
Introduction to the derivative of e^x, ln x, sin x, cos x, and tan x.

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Calculus: Quotient Rule and Simplifying
The quotient rule is useful when trying to find the derivative of a function that is divided by another function. As long as both functions have derivatives, the quotient rule tells us that the final derivative is a specific combination of both of the original functions and their derivatives.

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