# How do you differentiate f(x)=tan((e^x)^2) using the chain rule?

Jun 29, 2018

$\frac{\mathrm{df}}{\mathrm{dx}} = 2 {\left({e}^{x}\right)}^{2} {\sec}^{2} x$

#### Explanation:

$\frac{\mathrm{df}}{\mathrm{dx}} = \frac{d}{\mathrm{dx}} \left[\tan \left({\left({e}^{x}\right)}^{2}\right)\right]$

Chain rule:
$\frac{\mathrm{df}}{\mathrm{dx}} = {\sec}^{2} x \frac{d}{\mathrm{dx}} \left[{\left({e}^{x}\right)}^{2}\right]$

Note that ${\left({e}^{x}\right)}^{2} = {e}^{2 x}$:
$\frac{\mathrm{df}}{\mathrm{dx}} = {\sec}^{2} x \frac{d}{\mathrm{dx}} \left[{e}^{2 x}\right]$
$\frac{\mathrm{df}}{\mathrm{dx}} = 2 {e}^{2 x} {\sec}^{2} x$

Express it in the same fashion as the question:
$\frac{\mathrm{df}}{\mathrm{dx}} = 2 {\left({e}^{x}\right)}^{2} {\sec}^{2} x$