# How do you differentiate f(x)=e^tan(1/x)  using the chain rule?

Apr 14, 2016

$\frac{d}{\mathrm{dx}} \left({e}^{\tan} \left(\frac{1}{x}\right)\right) = {e}^{\tan} \left(\frac{1}{x}\right) \left[\frac{d}{\mathrm{dx}} \left(\tan \left(\frac{1}{x}\right)\right)\right]$
$= {e}^{\tan} \left(\frac{1}{x}\right) \left[{\sec}^{2} \left(\frac{1}{x}\right) \frac{d}{\mathrm{dx}} \left(\frac{1}{x}\right)\right]$
$= {e}^{\tan} \left(\frac{1}{x}\right) {\sec}^{2} \left(\frac{1}{x}\right) \left[\left(- \frac{1}{x} ^ 2\right)\right]$
$= - \frac{{e}^{\tan} \left(\frac{1}{x}\right) {\sec}^{2} \left(\frac{1}{x}\right)}{x} ^ 2$