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Method of Differentiation: Function 2xy

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

Question
If yy is a function of xx and log⁡(x+y)=2xy\log(x+y) = 2xy, then y′(0)y'(0) is.
Solution
Answer: 1
Step 1: Find y(0)y(0) At x=0x = 0: log⁡y=0⇒y=1\log y = 0 \Rightarrow y = 1. Step 2: Differentiate and evaluate at (0,1)(0, 1)
1x+y(1+dydx)=2(xdydx+y)\frac{1}{x+y}\left(1+\frac{dy}{dx}\right) = 2\left(x\frac{dy}{dx}+y\right)
 At (0,1)(0, 1): 1+y′=2⇒y′(0)=11+y' = 2 \Rightarrow y'(0) = 1. Answer: 1
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