Method of DifferentiationmediumFree

Method of Differentiation: Question

JEE Maths question with a full step-by-step solution.

Question
If xy=(x+y)nxy = (x+y)^n and dydx=yx\dfrac{dy}{dx} = \dfrac{y}{x}, then nn is.
Solution
Answer: 2
Step 1: Take logarithms and differentiate
log⁡x+log⁡y=nlog⁡(x+y)  ⟹  nx+y(1+dydx)=1x+1ydydx\log x + \log y = n\log(x+y) \implies \frac{n}{x+y}\left(1+\frac{dy}{dx}\right) = \frac{1}{x}+\frac{1}{y}\frac{dy}{dx}
 Step 2: Substitute dydx=yx\dfrac{dy}{dx} = \dfrac{y}{x}
nx+y⋅x+yx=1x+1x  ⟹  nx=2x  ⟹  n=2\frac{n}{x+y}\cdot\frac{x+y}{x} = \frac{1}{x}+\frac{1}{x} \implies \frac{n}{x} = \frac{2}{x} \implies n = 2
 Answer: 2
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