CBSE (Science) Class 12CBSE
Share
Notifications

View all notifications
Books Shortlist
Your shortlist is empty

Differentiate ( Tan X ) 1 / X ? - CBSE (Science) Class 12 - Mathematics

Login
Create free account


      Forgot password?

Question

Differentiate \[\left( \tan x \right)^{1/x}\] ?

Solution

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

  Is there an error in this question or solution?

Video TutorialsVIEW ALL [1]

Solution Differentiate ( Tan X ) 1 / X ? Concept: Simple Problems on Applications of Derivatives.
S
View in app×