The product rule is a formula that is used to find the derivative of the product of two or more functions. Given two differentiable functions, f(x) and g(x), where f'(x) and g'(x) are their respective derivatives, the product rule can be stated as,
or using abbreviated notation:
The product rule can be expanded for more functions. For example, for the product of three functions, f, g, and h, the product rule is:
It can be expanded in this same way for as many products of functions as necessary. Below are some examples of use of the product rule for different cases.
Use the product rule to find the following derivatives.
1. y = exsin(x)
Let f(x) = ex and g(x) = sin(x), then apply the product rule:
2. y = x3cos(x)ln(x)
Let f(x) = x3, g(x) = cos(x), and h(x) = ln(x), then apply the product rule:
Note that the product rule, like the quotient 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 quotient rule and chain rule in addition to the product rule; refer to the quotient and chain rule pages for more information on the rules.
Differentiate the following functions.
In order to differentiate this, we need to use both the product and the quotient rules. Given two differentiable functions f(x) and g(x) where g(x) ≠ 0, the quotient rule can be written as:
In this case, f(x) = x3tan(x), and g(x) = ex. Noting that we must use the product rule to differentiate f(x), y can be differentiated using both product and quotient rules as follows:
To find the derivative, we need to use both the product rule and the chain rule; the chain rule is used to differentiate sin(2x + 1).