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Class 12 Continuity And Differentiability · Long Answer3 Marksimplicit_differentiation

Class 12 Maths · Continuity And Differentiability · Long Answer #69

If (cos x)^y=(cos y)^x on a differentiable real branch with cos x>0,cos y>0 and ln(cos x)+x tan y≠0, find y′.

✅ Step-by-Step Solution

Final Answer: [ln(cos y)+y tan x]/[ln(cos x)+x tan y]
  1. Take logarithms to get y ln(cos x)=x ln(cos y).
  2. Differentiate by product and chain rules: y′ln(cos x)−y tan x=ln(cos y)−x tan y·y′.
  3. Collect terms: [ln(cos x)+x tan y]y′=ln(cos y)+y tan x.
  4. Dividing by the stated nonzero coefficient gives y′=[ln(cos y)+y tan x]/[ln(cos x)+x tan y].

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Take logarithms to get y ln(cos x)=x ln(cos y).
  • [1 mark] Differentiate by product and chain rules: y′ln(cos x)−y tan x=ln(cos y)−x tan y·y′.
  • [1 mark] Collect terms: [ln(cos x)+x tan y]y′=ln(cos y)+y tan x.
  • [0.5 marks] Dividing by the stated nonzero coefficient gives y′=[ln(cos y)+y tan x]/[ln(cos x)+x tan y].

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