What is dy/dx euqal to ?

Correct Answer is : 39−x2{3}{{9-x²}}9−x23

Correct Answer is : 39−x2{3}{{9-x²}}9−x23

Test Index

UPSC NDA I 2025 Mathematics Paper

Question Numbers: 91-92

Consider the following for the two (02) items that follow:

Let

y=sin^{-1}(x-\frac{4x^{3}}{27}).

Question : 92 of 120

Marks: +1, -0

What is

\frac{dy}{dx}

euqal to ?

Solution:

Concept:

The derivative of inverse sine is

\frac{d}{dx}\sin^{-1}(u)=\frac{1}{\sqrt{1-u^{2}}}\cdot\frac{du}{dx}

Explanation:

Given

y=3\sin^{-1}\left(\frac{x}{3}\right)

. Set

u=\frac{x}{3}

, so

\frac{du}{dx}=\frac{1}{3}

Using chain rule:

\frac{dy}{dx}=3\times\frac{1}{\sqrt{1-u^{2}}}\times\frac{du}{dx}=3\times\frac{1}{\sqrt{1-\left(\frac{x}{3}\right)^{2}}}\times\frac{1}{3}

Simplify:

\frac{dy}{dx}=\frac{1}{\sqrt{1-\frac{x^{2}}{9}}}=\frac{1}{\sqrt{\frac{9-x^{2}}{9}}}=\frac{3}{\sqrt{9-x^{2}}}

Answer:

Option C:

\frac{3}{\sqrt{9-x^{2}}}

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