# How do you differentiate f(x)= e^x(4x-4) using the product rule?

Jun 28, 2017

$\frac{\mathrm{df}}{\mathrm{dx}} = 4 x {e}^{x}$

#### Explanation:

Product rule states if $f \left(x\right) = g \left(x\right) h \left(x\right)$

then $\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{dg}}{\mathrm{dx}} \times h \left(x\right) + \frac{\mathrm{dh}}{\mathrm{dx}} \times g \left(x\right)$

Here we have $f \left(x\right) = g \left(x\right) h \left(x\right)$, where $g \left(x\right) = {e}^{x}$ and $h \left(x\right) = 4 x - 4$

Therefore $\frac{\mathrm{dg}}{\mathrm{dx}} = {e}^{x}$ and $\frac{\mathrm{dh}}{\mathrm{dx}} = 4$

Hence $\frac{\mathrm{df}}{\mathrm{dx}} = {e}^{x} \left(4 x - 4\right) + 4 {e}^{x}$

= ${e}^{x} \left(4 x - 4 + 4\right)$

= $4 x {e}^{x}$