# How do you find the derivative of f(x)=(2-4e^(2x))^3 ?

Nov 21, 2017

$- 24 {\left(2 - 4 {e}^{2 x}\right)}^{2} \cdot {e}^{2 x}$

#### Explanation:

This problem is best solved by a double application of the chain rule

$\frac{d}{\mathrm{dx}} \left({\left(2 - 4 {e}^{2 x}\right)}^{3}\right) =$

Let $u = 2 - 4 {e}^{2 x}$:

$= \frac{d}{\mathrm{du}} \left({u}^{3}\right) \frac{d}{\mathrm{dx}} \left(2 - 4 {e}^{2 x}\right) =$

Let $z = 2 x$:

$= 3 {u}^{2} \cdot \frac{d}{\mathrm{dz}} \left(2 - 4 {e}^{z}\right) \frac{d}{\mathrm{dx}} \left(2 x\right) =$

$= 3 {u}^{2} \cdot - 4 {e}^{z} \cdot 2 = - 24 {u}^{2} \cdot {e}^{z} =$

Resubstitute:

$= - 24 {\left(2 - 4 {e}^{2 x}\right)}^{2} \cdot {e}^{2 x}$