Question

Find the derivatives of the function: $f\left(x\right)=\ln \left({\tan }^2\left(x\right)\right).$

Short answer

The derivative of $f\left(x\right)=\ln \left({\tan }^2\left(x\right)\right)is\frac{2}{\sin \left(x\right)\cos \left(x\right)}.$

Step-by-step solution

Given information   

The given expression is $f\left(x\right)=\ln \left({\tan }^2\left(x\right)\right).$

Simplification     

$$\begin{gathered} f\text{'}\left(x\right) \\ =\frac{d}{dx}(\ln \left({\tan }^2\left(x\right)\right) \\ =\frac{1}{{\tan }^2\left(x\right)}\times 2\tan \left(x\right)se{c}^2\left(x\right) \\ =\frac{2}{\sin \left(x\right)\cos \left(x\right)} \\ Hence,f\text{'}(x)=\frac{2}{\sin \left(x\right)\cos \left(x\right)}. \end{gathered}$$