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Class 12 Continuity And Differentiability · Long Answer3 Marksimplicit_differentiation

Class 12 Maths · Continuity And Differentiability · Long Answer #131

If xy=e^(x−y) on a differentiable real branch, show y′=y(x−1)/[x(y+1)] wherever the denominator is nonzero.

✅ Step-by-Step Solution

Final Answer: y(x−1)/[x(y+1)]
  1. Implicit differentiation gives y+xy′=e^(x−y)(1−y′).
  2. Use e^(x−y)=xy to rewrite this as y+xy′=xy−xyy′.
  3. Collect derivatives: x(1+y)y′=y(x−1).
  4. Divide by x(1+y)≠0 to obtain the required formula.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Implicit differentiation gives y+xy′=e^(x−y)(1−y′).
  • [1 mark] Use e^(x−y)=xy to rewrite this as y+xy′=xy−xyy′.
  • [1 mark] Collect derivatives: x(1+y)y′=y(x−1).
  • [0.5 marks] Divide by x(1+y)≠0 to obtain the required formula.

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