If y= ≤ft(2 ^{-1} x) then d y/d x=..

Correct Answer is : 2−4x21−x2{2-4 x²}{{1-x²}}1−x22−4x2

Correct Answer is : 2−4x21−x2{2-4 x²}{{1-x²}}1−x22−4x2

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MHTCET 19 Apr 2026 Shift 1 Paper

Question : 121 of 150

Marks: +1, -0

y=\sin \left(2 \sin ^{-1} x\right)

\frac{d y}{d x}=.

Solution:

Concept:

Use the trigonometric identity

\sin 2\theta = 2\sin\theta \cos\theta

Substitute

\theta = \sin^{-1}x

and then differentiate the resulting product.

Explanation:

Let

\theta = \sin^{-1}x

Then

\sin\theta = x

and

\cos\theta = \sqrt{1-x^2}

y = \sin(2\theta) = 2\sin\theta \cos\theta = 2x\sqrt{1-x^2}

Differentiate

y = 2x\sqrt{1-x^2}

using the product rule.

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

Now

\frac{d}{dx}\left(\sqrt{1-x^2}\right) = \frac{-x}{\sqrt{1-x^2}}

Therefore

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

Combine the terms over a common denominator:

\frac{dy}{dx} = \frac{2(1-x^2) - 2x^2}{\sqrt{1-x^2}} = \frac{2 - 4x^2}{\sqrt{1-x^2}}

Answer:

\frac{dy}{dx} = \frac{2 - 4x^2}{\sqrt{1-x^2}}

, which matches Option A.

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