If y= ≤ft( ^{-1} x), then d y/d x at x=1 is equal to

Correct Answer is : 12{1}{{2}}21

Correct Answer is : 12{1}{{2}}21

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MHTCET 16 May 2024 Shift 1 Paper

Question : 131 of 150

Marks: +1, -0

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

\frac{d y}{d x}

x=1

is equal to

Solution:

Concept:

Use chain rule for differentiating a composite trigonometric function and simplify using

\tan^{-1}x

Explanation:

Given

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

Let

\theta=\tan^{-1}x

, so

\tan\theta=x

and

\sec\theta=\sqrt{1+x^2}

Differentiate

y=\sec\theta

with respect to

x

using the chain rule:

\frac{dy}{dx}=\sec\theta\tan\theta\cdot\frac{d\theta}{dx}

Here,

\frac{d}{dx}(\tan^{-1}x)=\frac{1}{1+x^2}

So,

\frac{dy}{dx}=\sqrt{1+x^2}\cdot x\cdot\frac{1}{1+x^2}

Simplify:

\frac{dy}{dx}=\frac{x}{\sqrt{1+x^2}}

Now substitute

x=1

\left(\frac{dy}{dx}\right)_{x=1}=\frac{1}{\sqrt{1+1^2}}=\frac{1}{\sqrt{2}}

Answer:

\frac{dy}{dx}

x=1

\frac{1}{\sqrt{2}}

Therefore, the correct option is B.

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