Product rule
The product rule is a formula that is used to determine the derivative of a product of functions. There are a few different ways that the product rule can be represented. Below is one of them. Given the product of two functions, f(x)g(x), the derivative of the product of those two functions can be denoted as (f(x)·g(x))'. Another way that the derivative is denoted is . The function does not necessarily have to be a function of x, so these are often abbreviated as (f · g)' or , where the apostrophe (') indicates a derivative:
(f · g)' = f ' · g + f · g'
Example
Given:
f(x) | = | x^{2} |
g(x) | = | 5x^{3} |
Plug f(x) and g(x) into the product rule formula:
[f(x) · g(x)]' | = | f'(x) · g(x) + f(x) · g'(x) |
(x^{2} · 5x^{3})' | = | 2x · 5x^{3} + x^{2} · 15x^{2} |
= | 10x^{4} + 15x^{4} | |
= | 25x^{4} |
This example uses only the power rule of derivatives for simplicity, but the product rule can be used with the numerous other derivative rules. The derivatives page has a table of derivative rules for your reference.
See also derivatives, quotient rule, chain rule.