# If f(x)= sec2 x  and g(x) = -x^2 -1 , how do you differentiate f(g(x))  using the chain rule?

Jan 7, 2016

First, let's find our new function: $f \left(g \left(x\right)\right) = \sec \left(- 2 {x}^{2} - 2\right)$

#### Explanation:

Now, we'll need chain rule to differentiate it.

• Chain rule: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}}$

Let's rename $u = - 2 {x}^{2} - 2$ and recall the rule to differentiate $\sec$ functions:

• Be $y = \sec u$, then $y ' = u ' \sec u \tan u$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - 4 x \sec \left(- 2 {x}^{2} - 2\right) \tan \left(- 2 {x}^{2} - 2\right)$