How do you differentiate f(x)=-xe^x*(4-x)/6 using the product rule?

Mar 20, 2018

$f ' \left(x\right) = \frac{x {e}^{x}}{6} - \frac{x {e}^{x} \left(4 - x\right)}{6} - \frac{{e}^{x} \left(4 - x\right)}{6}$

Explanation:

If a function $f \left(x\right) = u v w$ where $u$, $v$ and $w$ are all functions of $x$, then $f ' \left(x\right) = u v w ' + u v ' w + u ' v w$

$u = - x$
$u ' = - 1$

$v = {e}^{x}$
$v ' = {e}^{x}$

$w = \frac{4 - x}{6} = \frac{4}{6} - \frac{x}{6}$
$w ' = - \frac{1}{6}$

$f ' \left(x\right) = \frac{x {e}^{x}}{6} - \frac{x {e}^{x} \left(4 - x\right)}{6} - \frac{{e}^{x} \left(4 - x\right)}{6}$