Question

Differentiate

${\left(\tan x\right)}^{1/x}$

Answer 1

$$\begin{gathered} \mathrm{Let}y={\left(\tan x\right)}^{\frac{1}{x}}...\left(i\right) \\ \mathrm{Taking}\log \mathrm{on}\mathrm{both}\mathrm{sides}, \\ \log y=\log {\left(\tan x\right)}^{\frac{1}{x}} \\ \Rightarrow \log y=\frac{1}{x}\log \left(\tan x\right) \\ \mathrm{Differentiating}\mathrm{with}\mathrm{respect}\mathrm{to}x, \\ \frac{1}{y}\frac{dy}{dx}=\frac{1}{x}\frac{d}{dx}\left\{\log \left(\tan x\right)\right\}+\log \left(\tan x\right)\frac{d}{dx}\left(\frac{1}{x}\right) \\ \Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{1}{x}\times \frac{1}{\tan x}\frac{d}{dx}\left(\tan x\right)+\log \left(\tan x\right)\left(-\frac{1}{x^2}\right) \\ \Rightarrow \frac{1}{y}\frac{dy}{dx}=\frac{1}{x\tan x}\left({\sec }^{2}x\right)-\frac{\log \left(\tan x\right)}{x^2} \\ \Rightarrow \frac{dy}{dx}=y\left[\frac{{\sec }^{2}x}{x\tan x}-\frac{\log \left(\tan x\right)}{x^2}\right] \\ \Rightarrow \frac{dy}{dx}={\left(\tan x\right)}^{\frac{1}{x}}\left[\frac{{\sec }^{2}x}{x\tan x}-\frac{\log \left(\tan x\right)}{x^2}\right]\left[\mathrm{using}\mathrm{equation}\left(\mathrm{i}\right)\right] \end{gathered}$$