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Class 12 Continuity And Differentiability · Long Answer3 Marksimplicit_differentiation

Class 12 Maths · Continuity And Differentiability · Long Answer #66

If arctan(y/x)=ln√(x²+y²), prove y′=(x+y)/(x−y), where x≠0,x≠y and the relation defines a differentiable branch.

✅ Step-by-Step Solution

Final Answer: (x+y)/(x−y)
  1. Rewrite the logarithm as ½ln(x²+y²).
  2. Implicit differentiation gives (xy′−y)/(x²+y²)=(x+yy′)/(x²+y²).
  3. Cancel the positive denominator and collect: (x−y)y′=x+y.
  4. Divide by x−y≠0 to obtain y′=(x+y)/(x−y).

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Rewrite the logarithm as ½ln(x²+y²).
  • [1 mark] Implicit differentiation gives (xy′−y)/(x²+y²)=(x+yy′)/(x²+y²).
  • [1 mark] Cancel the positive denominator and collect: (x−y)y′=x+y.
  • [0.5 marks] Divide by x−y≠0 to obtain y′=(x+y)/(x−y).

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