Derivative Product Rule
Simple overview of the derivative product rule.
Main Idea
We use the derivative product rule when the function we are differentiating is the product of two separate functions.
\[f(x)=g(x)h(x)\]The derivative of this product is then
\[f'(x)=g(x)h'(x)+h(x)g'(x)\]“The product of the first function time the derivative of the second function plus the second function times the derivative of the first function.”
Watch this section: introduction at 0:00.
Example 1
Consider
\[f(x)=x^2\cos(x)\]We can think of $f(x)$ as the product of the following two functions.
\[g(x)=x^2 \qquad\text{and}\qquad h(x)=\cos(x)\]Their derivatives are
\[g'(x)=2x \qquad\text{and}\qquad h'(x)=-\sin(x)\]Now place $f(x)$, $g(x)$, $f’(x)$, and $g’(x)$ into the product rule (\(f'(x)=g(x)h'(x)+h(x)g'(x)\)).
\[f'(x)=x^2\left(-\sin(x)\right)+\cos(x)(2x)\]Watch this section: derivative of $x^2\cos(x)$ at 0:35.
Example 2
Consider
\[f(x)=e^x(x^4+1)\]The two functions are
\[g(x)=e^x \qquad\text{and}\qquad h(x)=x^4+1\]Differentiate each function
\[g'(x)=e^x \qquad\text{and}\qquad h'(x)=4x^3\]Then use the product rule
\[f'(x)=e^x(4x^3)+(x^4+1)e^x\]Watch this section: derivative of $e^x(x^4+1)$ at 1:55.
Timestamp Guide
Related Calculus Work
Use the Product Rule topic page for more product-rule material.
For nearby derivative rules, review the Power Rule, Chain Rule, and Quotient Rule.