Question

Find the derivative of ${\left[\ln \right(\tan x\left)\right]}^2$.

Answer 1

Find the required derivative.

Given function: ${\left[\ln \right(\tan x\left)\right]}^2$.

Differentiating with respect to $x$, we get,

$$\begin{gathered} \frac{d}{dx}{\left[\ln \right(\tan x\left)\right]}^2=2\ln \left(\tan x\right)\frac{d}{dx}\ln \left(\tan x\right)\left[\because \frac{d}{dx}{\left(f\left(x\right)\right)}^n=n{\left(f\left(x\right)\right)}^{n-1}\frac{d}{dx}f\left(x\right)\right] \\ =2\ln \left(\tan x\right)\times \frac{1}{\tan x}\times {\sec }^2\mathrm{x}\left[\because \frac{d}{dx}\ln \left(f\left(x\right)\right)=\frac{1}{f\left(x\right)}\frac{d}{dx}f\left(x\right)\right] \\ =2\ln \left(\tan x\right)\times \frac{\cos x}{\sin x}\times \frac{1}{{\left(\cos x\right)}^2} \\ =2\ln \left(\tan x\right)\times \frac{1}{\sin x}\times \frac{1}{\left(\cos x\right)} \\ =2\ln \left(\tan x\right)\mathrm{cosec}x\sec x \end{gathered}$$

Hence, the derivative of ${\left[\ln \right(\tan x\left)\right]}^2$is $2\ln \left(\tan x\right)\mathrm{cosec}x\sec x$.