What is d y/d x equal to, given that y=y₀ when x=1 ?

Correct Answer is : y0ey01−ey0{y₀ e^{y₀}}{1-e^{y₀}}1−ey0y0ey0

Correct Answer is : y0ey01−ey0{y₀ e^{y₀}}{1-e^{y₀}}1−ey0y0ey0

Test Index

UPSC NDA I 2026 Mathematics Paper

Question Numbers: 81-82

For the next two (02) items that follow:

Let

\left(e^{y}\right)^{x}-y=0

, where

y

is a function of

x

whose domain is

(0,10]

Question : 82 of 120

Marks: +1, -0

What is

\frac{d y}{d x}

y=y_{0}

x=1

Solution:

Concept:

Differentiate the given implicit relation and substitute the given point condition.

Explanation:

Given:

(e^y)^x - y = 0

, i.e.

e^{xy} - y = 0

Differentiating with respect to

x

e^{xy}\left(y+x\frac{dy}{dx}\right)-\frac{dy}{dx}=0

\frac{dy}{dx}=\frac{y e^{xy}}{1-x e^{xy}}

x=1

y=y_0

. Therefore:

\frac{dy}{dx}=\frac{y_0 e^{y_0}}{1-e^{y_0}}

This matches Option D.

Answer:

Option D.

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