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Class 12 Continuity And Differentiability · Short Answer2 Marksimplicit_differentiation

Class 12 Maths · Continuity And Differentiability · Short Answer #56

Assume the positive real infinite power tower y=(cos x)^((cos x)^(…)) converges to a differentiable limit, with cos x>0 and y ln(cos x)≠1. Show y′=y²tan x/[y ln(cos x)−1].

✅ Step-by-Step Solution

Final Answer: y′=y²tan x/[y ln(cos x)−1]
  1. A convergent tower satisfies y=(cos x)^y, so ln y=y ln(cos x).
  2. Differentiate: y′/y=y′ln(cos x)−y tan x. Multiply by y and collect: [1−y ln(cos x)]y′=−y²tan x.
  3. Dividing by the nonzero coefficient gives the claimed formula.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] A convergent tower satisfies y=(cos x)^y, so ln y=y ln(cos x).
  • [1 mark] Differentiate: y′/y=y′ln(cos x)−y tan x. Multiply by y and collect: [1−y ln(cos x)]y′=−y²tan x.
  • [0.5 marks] Dividing by the nonzero coefficient gives the claimed formula.

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