How do you differentiate xe^x?

Mar 18, 2016

Use the product rule.

Explanation:

The Product rule (for derivatives) says that for differentiable functions, $f$ and $g$, the derivative of the product is given by:
$\frac{d}{\mathrm{dx}} \left(f \left(x\right) g \left(x\right)\right) = f ' \left(x\right) g \left(x\right) + f \left(x\right) g ' \left(x\right)$

In this question, we have

$f \left(x\right) = x$, so $f ' \left(x\right) = 1$, and

$g \left(x\right) = {e}^{x}$, so $g ' \left(x\right) = {e}^{x}$

$\frac{d}{\mathrm{dx}} \left(f \left(x\right) g \left(x\right)\right) = \left(1\right) \left({e}^{x}\right) + x \left({e}^{x}\right)$

$= {e}^{x} + x {e}^{x}$ $\text{ }$ which you may prefer to write as

$= {e}^{x} \left(1 + x\right)$ or as ${e}^{x} \left(x + 1\right)$ or in some other way.