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

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ConceptSimple Problems on Applications of Derivatives

#### Question

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

#### Solution

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

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Solution Differentiate ( Tan X ) 1 / X ? Concept: Simple Problems on Applications of Derivatives.
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