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Class 12 Continuity And Differentiability · Long Answer3 Marksimplicit_differentiation

Class 12 Maths · Continuity And Differentiability · Long Answer #85

On a differentiable real branch where the powers are real and x,y,x+y and nx−my are nonzero, let x^m y^n=C(x+y)^(m+n), with fixed nonzero constants m,n,C. Prove dy/dx=y/x.

✅ Step-by-Step Solution

Final Answer: y′=y/x
  1. Take logarithms of absolute values: m ln|x|+n ln|y|=ln|C|+(m+n)ln|x+y|.
  2. Differentiate: m/x+(n/y)y′=(m+n)(1+y′)/(x+y).
  3. Collect the y′ terms to get [n/y−(m+n)/(x+y)]y′=(m+n)/(x+y)−m/x. The two coefficients simplify to (nx−my)/[y(x+y)] and (nx−my)/[x(x+y)] respectively.
  4. Where nx−my≠0 their ratio is y/x, proving the claim for regular branches.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Take logarithms of absolute values: m ln|x|+n ln|y|=ln|C|+(m+n)ln|x+y|.
  • [1 mark] Differentiate: m/x+(n/y)y′=(m+n)(1+y′)/(x+y).
  • [1 mark] Collect the y′ terms to get [n/y−(m+n)/(x+y)]y′=(m+n)/(x+y)−m/x. The two coefficients simplify to (nx−my)/[y(x+y)] and (nx−my)/[x(x+y)] respectively.
  • [0.5 marks] Where nx−my≠0 their ratio is y/x, proving the claim for regular branches.

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