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NDA 2026
Exam
NDA
Year
2026
Subject
Mathematics
Chapter
Differentiation

NDA 2026 · Mathematics · Differentiation

Passage: Let (e^y)^x-y=0 , where y is a function of x whose domain...

Passage: Let (ey)x−y=0\left(e^y\right)^x-y=0, where yy is a function of xx whose domain is (0,10](0,10].

Question: What is dydx\dfrac{dy}{dx} equal to ?

Options

  1. Ay1−xy\dfrac{y}{1-xy}
  2. By1+xy\dfrac{y}{1+xy}
  3. Cy21−xy\dfrac{y^2}{1-xy}
  4. Dy21+xy\dfrac{y^2}{1+xy}

Correct answer

C. y21−xy\dfrac{y^2}{1-xy}

Step-by-step solution

Given equation is (ey)x−y=0(e^y)^x - y = 0

exy=ye^{xy} = y

Taking natural logarithm on both sides:

xy=ln⁡(y)xy = \ln(y)

Differentiating both sides with respect to xx:

y+xdydx=1ydydxy + x \dfrac{dy}{dx} = \dfrac{1}{y} \dfrac{dy}{dx}

y=(1y−x)dydxy = \left( \dfrac{1}{y} - x \right) \dfrac{dy}{dx}

y=(1−xyy)dydxy = \left( \dfrac{1 - xy}{y} \right) \dfrac{dy}{dx}

dydx=y21−xy\dfrac{dy}{dx} = \dfrac{y^2}{1 - xy}
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