QJEEM13Ma26fc: JEE Main 2013 Mathematics - Class 12 JEEM13Ma26fc Solution: If y = sec ⁡ ( tan − 1 x ) , then d y d x at x = 1 is equal to :

Question

Verified Solution • 2025–26 Session moderate

If

y = \sec (\tan^{- 1} x),

then

\frac{d y}{d x}

at

x = 1

is equal to :

Accepted Answer

Let

y = \sec (\tan^{- 1} x)

and

\tan^{- 1} x = θ .

\Rightarrow x = \tan θ

Thus, we have

y = \sec θ

\Rightarrow y = \sqrt{1 + x^{2}}

(

As

\sec^{2} θ = 1 + \tan^{2} θ

)

\Rightarrow \frac{d y}{d x} = \frac{1}{2 \sqrt{1 + x^{2}}} .2 x

At

x = 1, \frac{d y}{d x} = \frac{1}{\sqrt{2}} .

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