All courses › Mathematics R1 › The product rule

The product rule

The product rule differentiates a product of two functions. You differentiate one and leave the other, then add the reverse. For $xe^x$ both terms contain a factor $e^x$, which can be taken outside the brackets. The same rule applies when one factor is a logarithm. Differentiate each factor on its own before you put them back together.

Symbols

Example

$(xe^x)' = e^x + xe^x = (1 + x)e^x$, so the derivative at $x = 0$ is $1$. $(x\ln x)' = \ln x + 1$, which is $2$ when $x = e$.

← The chain rule · Maxima and minima →

Part of Mathematics R1: Differentiation and curve analysis.