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# Differential Coefficient of Sec Sec ( Tan − 1 X ) is - CBSE (Science) Class 12 - Mathematics

ConceptSimple Problems on Applications of Derivatives

#### Question

Differential coefficient of sec $\sec \left( \tan^{- 1} x \right)$ is _____________ .

• $\frac{x}{1 + x^2}$

• $x \sqrt{1 + x^2}$

• $\frac{1}{\sqrt{1 + x^2}}$

• $\frac{x}{\sqrt{1 + x^2}}$

#### Solution

$\frac{x}{\sqrt{1 + x^2}}$

$\text{We have, y } = \sec\left( \tan^{- 1} x \right)$
$\Rightarrow \frac{dy}{dx} = \sec\left( \tan^{- 1} x \right) \tan\left( \tan^{- 1} x \right) \times \frac{d}{dx}\left( \tan^{- 1} x \right)$
$\Rightarrow \frac{dy}{dx} = \sec\left( \tan^{- 1} x \right) \tan\left( \tan^{- 1} x \right) \times \frac{1}{\sqrt{1 + x^2}}$
$\Rightarrow \frac{dy}{dx} = y\left( \frac{x}{\sqrt{1 + x^2}} \right)$
$\Rightarrow \frac{dy}{dx} = \left( \frac{x}{\sqrt{1 + x^2}} \right) y$

This is the equation of differential equation which have coefficient $\frac{x}{\sqrt{1 + x^2}}$.

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Solution Differential Coefficient of Sec Sec ( Tan − 1 X ) is Concept: Simple Problems on Applications of Derivatives.
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