# How do you find the derivative of f (x) = x^6 · e^(7x)?

Jan 21, 2016

$f ' \left(x\right) = {x}^{5} {e}^{7 x} \left(7 x + 6\right)$

#### Explanation:

Use the product rule, which states that for a function $f \left(x\right) = g \left(x\right) h \left(x\right)$, the derivative is

$f ' \left(x\right) = g ' \left(x\right) h \left(x\right) + g \left(x\right) h ' \left(x\right)$

We have:

$g \left(x\right) = {x}^{6}$
$h \left(x\right) = {e}^{7 x}$

Find the derivatives of either function.

Through the power rule,

$g ' \left(x\right) = 6 {x}^{5}$

Through the chain rule, particularly applied to $h \left(x\right)$,

$h ' \left(x\right) = {e}^{7 x} \frac{d}{\mathrm{dx}} \left[7 x\right] = 7 {e}^{7 x}$

Plug both of these into the original equation.

$f ' \left(x\right) = 6 {x}^{5} {e}^{7 x} + {x}^{6} \left(7 {e}^{7 x}\right)$

$f ' \left(x\right) = {x}^{5} {e}^{7 x} \left(7 x + 6\right)$