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CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Q313 Markslogarithmic_differentiation

CBSE Class 12 Mathematics 2026 Main Paper 65/3/3 · Question 31 (Continuity And Differentiability)

If \((\sin x)^y=y^{\cos x}\), find dy/dx on branches with y>0 and sin x>0.

✅ Step-by-Step Solution

Final Answer: y′=y(y cot x+sin x·ln y)/(cos x−y ln(sin x)), where the denominator is nonzero.
  1. Take natural logs to obtain y ln(sin x)=cos x ln y.
  2. Differentiate implicitly: y′ ln(sin x)+y cot x=−sin x ln y+(cos x/y)y′.
  3. Collect y′ terms and divide to obtain the stated derivative.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Take natural logs to obtain y ln(sin x)=cos x ln y.
  • [2 marks] Differentiate implicitly: y′ ln(sin x)+y cot x=−sin x ln y+(cos x/y)y′.
  • [0.5 marks] Collect y′ terms and divide to obtain the stated derivative.

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