Method of DifferentiationeasyFree

Method of Differentiation: Function 2xy Find Value

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

Question
If yy is a function of xx and ln⁡(x+y)=2xy\ln(x+y) = 2xy, then find the value of y′′(0)y''(0).
Solution
Answer: 8
Step 1: Find y(0)y(0) At x=0x = 0: ln⁡y=0⇒y=1\ln y = 0 \Rightarrow y = 1. Step 2: First derivative at (0,1)(0, 1) Differentiating ln⁡(x+y)=2xy\ln(x+y) = 2xy with respect to xx:
1+y′x+y=2(xy′+y)\frac{1+y'}{x+y} = 2(xy'+y)
 At (0,1)(0, 1): 1+y′=2⇒y′=11+y' = 2 \Rightarrow y' = 1. Step 3: Second derivative at (0,1)(0, 1) Differentiating the above relation with respect to xx:
(x+y)y′′−(1+y′)2(x+y)2=2(xy′′+2y′)\frac{(x+y)y'' - (1+y')^2}{(x+y)^2} = 2(xy''+2y')
 Substituting x=0x = 0, y=1y = 1, y′=1y' = 1:
y′′−41=4⇒y′′=8\frac{y'' - 4}{1} = 4 \Rightarrow y'' = 8
 Answer: 8
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