Find the derivative of the function?

$y = \ln {\tan}^{2} 3 x$

Mar 8, 2018

$\frac{\mathrm{dy}}{\mathrm{dx}} = 12 \csc \left(6 x\right)$

Explanation:

We to find the derivative of

$y = \ln \left({\tan}^{2} \left(3 x\right)\right) = 2 \ln \left(\tan \left(3 x\right)\right)$

Use the chain rule, if $y = f \left(u\right)$ and $u = g \left(x\right)$ then

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{\mathrm{dy}}{\mathrm{du}} \frac{\mathrm{du}}{\mathrm{dx}}$

Let $y = 2 \ln \left(u\right) \implies \frac{\mathrm{dy}}{\mathrm{du}} = \frac{2}{u}$

and $u = \tan \left(3 x\right) \implies \frac{\mathrm{du}}{\mathrm{dx}} = 3 {\sec}^{2} \left(3 x\right)$

Thus

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6}{u} {\sec}^{2} \left(3 x\right) = \frac{6}{\tan} \left(3 x\right) {\sec}^{2} \left(3 x\right) = 6 \csc \left(3 x\right) \sec \left(3 x\right)$

Or by applying the identity $\csc \left(2 x\right) = \frac{1}{2} \csc \left(x\right) \sec \left(x\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = 12 \csc \left(6 x\right)$