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

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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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