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Class 12 Continuity And Differentiability · Long Answer3 Marksimplicit_differentiation

Class 12 Maths · Continuity And Differentiability · Long Answer #145

On a differentiable real branch of x^m y^n=C(x+y)^(m+n), with fixed nonzero constants m,n,C, powers real, and x,y,x+y,nx−my all nonzero, prove y″=0.

✅ Step-by-Step Solution

Final Answer: y″=0
  1. Logarithmic differentiation of absolute values gives m/x+(n/y)y′=(m+n)(1+y′)/(x+y).
  2. Collecting gives [(nx−my)/(y(x+y))]y′=(nx−my)/(x(x+y)), hence y′=y/x under the stated nonzero conditions.
  3. Differentiate y/x by the quotient rule: y″=(xy′−y)/x²=(x(y/x)−y)/x²=0.
  4. Therefore y″=0 on every such regular branch.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Logarithmic differentiation of absolute values gives m/x+(n/y)y′=(m+n)(1+y′)/(x+y).
  • [1 mark] Collecting gives [(nx−my)/(y(x+y))]y′=(nx−my)/(x(x+y)), hence y′=y/x under the stated nonzero conditions.
  • [1 mark] Differentiate y/x by the quotient rule: y″=(xy′−y)/x²=(x(y/x)−y)/x²=0.
  • [0.5 marks] Therefore y″=0 on every such regular branch.

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