# If f(x)= tan8 x  and g(x) = 2 x , how do you differentiate f(g(x))  using the chain rule?

May 16, 2018

Derivative of $f \left(g \left(x\right)\right)$ is $32 {\sec}^{2} 16 x$.

#### Explanation:

As $f \left(x\right) = \tan 8 x$, $\frac{\mathrm{df}}{\mathrm{dx}} = {\sec}^{2} 8 x \cdot 8 = 16 {\sec}^{2} 8 x$

and as $g \left(x\right) = 2 x$ $\frac{\mathrm{dg}}{\mathrm{dx}} = 2$

and $f \left(g \left(x\right)\right) = \tan 8 \left(2 x\right) = \tan 16 x$

and according to chain rule

$\frac{\mathrm{df}}{\mathrm{dx}} = \frac{\mathrm{df}}{\mathrm{dg} \left(x\right)} \cdot \frac{\mathrm{dg}}{\mathrm{dx}}$

= $16 {\sec}^{2} 8 \left(g \left(x\right)\right) \cdot 2$

= $16 {\sec}^{2} \left(8 \cdot 2 x\right) \cdot 2$

= $32 {\sec}^{2} 16 x$