Question

What is the derivative of $\tan 2x$?

Answer 1

We know that the derivative of $\tan x$ is,

$\frac{d}{dx}\tan x={\sec }^{2}x$

Chain rule: $\frac{d}{dx}\left(fg\left(x\right)\right)=f\text{'}\left(g\left(x\right)\right)·g\text{'}\left(x\right)$

Differentiate $\tan 2x$ with respect to $x$, using the chain rule we get

$\frac{d}{dx}\tan 2x={\sec }^2\left(2x\right)\frac{d}{dx}\left(2x\right)$

$\Rightarrow$$\frac{d}{dx}\tan 2x={\sec }^2\left(2x\right)·2$

$\Rightarrow$$\frac{d}{dx}\tan 2x=2{\sec }^2\left(2x\right)$

Hence, the derivative of $\tan 2x$ is $2{\sec }^2\left(2x\right)$ .