How do you find the derivative of the function y = sin(tan(5x))?

May 24, 2016

$\setminus \cos \left(\setminus \tan \left(5 x\right)\right) \setminus {\sec}^{2} \left(5 x\right) 5$

Explanation:

$\setminus \frac{d}{\mathrm{dx}} \left(\setminus \sin \left(\setminus \tan \left(5 x\right)\right)\right)$

Applying chain rule,
$\setminus \frac{\mathrm{df} \left(u\right)}{\mathrm{dx}} = \setminus \frac{\mathrm{df}}{\mathrm{du}} \setminus \cdot \setminus \frac{\mathrm{du}}{\mathrm{dx}}$

Let $\tan \left(5 x\right)$ = u

We know,
$\setminus \frac{d}{\mathrm{du}} \left(\setminus \sin \left(u\right)\right) = \setminus \cos \left(u\right)$

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

So,
$= \setminus \cos \left(u\right) \setminus {\sec}^{2} \left(5 x\right) 5$

Substituting back,$\tan \left(5 x\right)$ = u

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