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

Apr 25, 2016

Multiply the derivative of ${e}^{\tan} x$ by the derivative of $\tan x$ to get $f ' \left(x\right) = {e}^{\tan x} {\sec}^{2} x$.

#### Explanation:

Differentiating this will require use of the chain rule, which, put plainly, states that the derivative of a composite function (like ${e}^{\tan} x$) is equal to the derivative of the "inside function" (in this case $\tan x$) multiplied by the derivative of the whole function (${e}^{\tan} x$).

In math terms, we say the derivative of the composite function $f \left(g \left(x\right)\right)$ is $f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$.

So, the derivative of ${e}^{\tan} x$ will be the derivative of ${e}^{\tan} x$, which is just ${e}^{\tan} x$ (the derivative of $e$ to the anything is $e$ to the anything) times the derivative of $\tan x$, which is ${\sec}^{2} x$. That is to say:
$\frac{d}{\mathrm{dx}} {e}^{\tan} x = {e}^{\tan} x \cdot \left(\tan x\right) ' = {e}^{\tan} x {\sec}^{2} x$