# Quotient rule

The quotient rule is a formula that is used to find the derivative of the quotient of two functions. Given two differentiable functions, f(x) and g(x), where f'(x) and g'(x) are their respective derivatives, the quotient rule can be stated as

or using abbreviated notation:

Examples

Use the quotient rule to find the following derivatives.

1.

Let f(x) = e^{x} and g(x) = 3x^{3}, then apply the quotient rule:

2.

Let f(x) = sin(x) and g(x) = x^{2}, then apply the quotient rule:

Note that the quotient rule, like the product rule, chain rule, and others, is simply a method of differentiation. It can be used on its own, or in combination with other methods. The following examples will use the product rule and chain rule in addition to the quotient rule; refer to the product or chain rule pages for more information on the rules.

Examples

Find the following derivatives.

1.

In order to differentiate this, we need to use both the quotient and product rule since the numerator involves a product of functions. Given two differentiable functions f(x) and g(x), the product rule can be written as:

Given the above, let f(x) = xe^{x} and g(x) = x + 2, then apply both the quotient and product rules:

2.

To differentiate this, we need to use both the quotient rule and the chain rule. Let f(x) = sin(x^{2}) and g(x) = (x^{3}+1)^{2} then apply the quotient and chain rules: