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MHTCET 18 Apr 2026 Shift 1 Paper

Question : 124 of 150

Marks: +1, -0

x e^{x y}=y+\sin ^{2} x

\frac{d y}{d x}

x=0

is equal to

Solution:

Concept:

Implicit differentiation and evaluating the derivative at a given point.

Explanation:

Given equation:

x e^{xy} = y + \sin^2 x

First, find

y

when

x = 0

Substitute

x = 0

0 \cdot e^{0} = y + \sin^2 0

0 = y + 0

So,

y = 0

Now differentiate both sides with respect to

x

Use the product rule on the left side:

\frac{d}{dx}(x e^{xy}) = e^{xy} + x e^{xy}\frac{d}{dx}(xy)

Since

\frac{d}{dx}(xy) = y + x\frac{dy}{dx}

, we get:

e^{xy} + x e^{xy}\left(y + x\frac{dy}{dx}\right) = \frac{dy}{dx} + 2\sin x \cos x

Now substitute

x = 0

and

y = 0

e^{0} + 0 = \frac{dy}{dx} + 2\sin 0 \cos 0

1 = \frac{dy}{dx} + 0

Thus,

\frac{dy}{dx} = 1

x = 0

Answer:

\frac{dy}{dx} = 1

Correct option: B.

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