# If f(x)= tan5 x  and g(x) = 2x^2 -1 , how do you differentiate f(g(x))  using the chain rule?

Jan 2, 2016

$20 x {\sec}^{2} \left(10 {x}^{2} - 5\right)$

#### Explanation:

The chain rule states that

$\frac{d}{\mathrm{dx}} \left[f \left(g \left(x\right)\right)\right] = f ' \left(g \left(x\right)\right) \cdot g ' \left(x\right)$

First, find $f ' \left(g \left(x\right)\right)$.

$f ' \left(x\right) = 5 {\sec}^{2} \left(5 x\right)$

Note that this required the chain rule as well.

$f ' \left(g \left(x\right)\right) = 5 {\sec}^{2} \left(5 \left(2 {x}^{2} - 1\right)\right) = 5 {\sec}^{2} \left(10 {x}^{2} - 5\right)$

Now, find $g ' \left(x\right)$.

$g ' \left(x\right) = 4 x$

Combine.

$\frac{d}{\mathrm{dx}} \left[f \left(g \left(x\right)\right)\right] = 5 {\sec}^{2} \left(10 {x}^{2} - 5\right) \cdot 4 x = 20 x {\sec}^{2} \left(10 {x}^{2} - 5\right)$