# How do you differentiate y=xe^x?

Jan 23, 2017

$\frac{\text{d"}{"d} x}{x {e}^{x}} = \left(1 + x\right) {e}^{x}$

Use the product rule.

#### Explanation:

The product rule:

If $u$ and $v$ are differentiable functions of $x$,

and $f = u \cdot v$,

then $f ' = u ' \cdot v + u \cdot v '$,

where the apostrophe denotes the derivative with respect to $x$.

In the above question, we can see that $x {e}^{x}$ is a product of $x$ and ${e}^{x}$, both which are elementary functions.

Thus

$\frac{\text{d"}{"d"x}(xe^x) = frac{"d"}{"d"x}(x) e^x + x frac{"d"}{"d} x}{{e}^{x}}$

$= \left(1\right) {e}^{x} + x \left({e}^{x}\right)$

$= \left(1 + x\right) {e}^{x}$