What is d y/d x equal to ?

Correct Answer is : y21−xy{y²}{1-x y}1−xyy2

Correct Answer is : y21−xy{y²}{1-x y}1−xyy2

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 : 81 of 120

Marks: +1, -0

What is

\frac{d y}{d x}

equal to ?

Solution:

Concept:

Use implicit differentiation after rewriting

(e^y)^x = e^{xy}

Explanation:

Given equation:

(e^y)^x - y = 0 \Rightarrow e^{xy} = y

Differentiate both sides with respect to

x

e^{xy}(y + x y') = y'

Since

e^{xy}=y

, substitute:

y(y + x y') = y'

y^2 + xy y' = y'

y'(1 - xy) = y^2

\frac{dy}{dx} = \frac{y^2}{1 - xy}

Answer:

\frac{y^{2}}{1-x y}

, i.e. option C.

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