# If f(x)= tan2 x  and g(x) = sqrt(-4x-3 , how do you differentiate f(g(x))  using the chain rule?

Mar 11, 2018

-(4sec^2(2sqrt(-4x-3))) /sqrt(-4x-3)

#### Explanation:

Starting with: $f \left(x\right) = \tan 2 x$ and $g \left(x\right) = \sqrt{- 4 x - 3} = {\left(- 4 x - 3\right)}^{\frac{1}{2}}$

Let $h \left(x\right) = f \left(g \left(x\right)\right) = \tan \left(2 \left(\sqrt{- 4 x - 3}\right)\right)$

$f ' \left(x\right) = 2 {\sec}^{2} \left(2 x\right)$
$g ' \left(x\right) = \left(\frac{1}{2}\right) \left(- 4\right) {\left(- 4 x - 3\right)}^{- \frac{1}{2}} = - \frac{2}{\sqrt{- 4 x - 3}}$

Using the Chain Rule:

$h ' \left(x\right) = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

$h ' \left(x\right) = 2 {\sec}^{2} \left(2 \sqrt{- 4 x - 3}\right) \cdot - \frac{2}{\sqrt{- 4 x - 3}}$

$h ' \left(x\right) = - \frac{4 {\sec}^{2} \left(2 \sqrt{- 4 x - 3}\right)}{\sqrt{- 4 x - 3}}$