# What is the derivative of y = sin(tan(5x))?

Dec 16, 2015

$y ' = 5 {\sec}^{2} \left(5 x\right) \cos \left(\tan \left(5 x\right)\right)$

#### Explanation:

Use the chain rule. The first overriding issue is the tangent function inside the sine function.

$y ' = \cos \left(\tan \left(5 x\right)\right) \frac{d}{\mathrm{dx}} \left[\tan \left(5 x\right)\right]$

Next, use chain rule again to find the derivative of the tangent function.

$\frac{d}{\mathrm{dx}} \left[\tan \left(5 x\right)\right] = {\sec}^{2} 5 x \frac{d}{\mathrm{dx}} \left[5 x\right] = 5 {\sec}^{2} \left(5 x\right)$

Multiply this back in to find $y '$.

$y ' = 5 {\sec}^{2} \left(5 x\right) \cos \left(\tan \left(5 x\right)\right)$