# What is the derivative of y= sin(tan 2x)?

Jan 5, 2016

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

#### Explanation:

We need to apply the chain rule twice.

Recall that the chain rule states, if we have some function $f \left(g \left(x\right)\right)$, the derivative of $f$ with respect to $x$ is equal to the derivative of $f$ with respect to $g$, multiplied by the derivative of $g$ with respect to $x$.

So in this case, the derivative $\frac{\mathrm{dy}}{\mathrm{dx}}$ will equal the derivative of $\sin \left(\tan 2 x\right)$ with respect to $\tan 2 x$ (basically, treat $\tan 2 x$ as a whole variable) times the derivative of $\tan 2 x$ with respect to $x$.

Derivative of $\sin$ is just $\cos$:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \cos \left(\tan 2 x\right) \cdot \frac{d}{\mathrm{dx}} \left[\tan 2 x\right]$

Derivative of $\tan$ is ${\sec}^{2}$. However, we need to apply the chain rule again, meaning this time we will just pull the derivative of $2 x$ out. (which is just $2$)

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