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CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Q273 Marksimplicit_derivatives

CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3 · Question 27 (Continuity And Differentiability)

The printed question sets \(y=\sqrt{\ln(\sin(x^2/3+1))}\) and asks to show \(y\prime=x\cot(x^2/3+1)/(3y)\). Derive the formal identity and examine its real domain.

✅ Step-by-Step Solution

Final Answer: The identity follows formally where a differentiable nonzero branch exists. The printed expression has no real differentiable interval.
  1. Write u=x²/3+1 and formally differentiate y²=ln(sin u) to get 2yy′=(2x/3)cot u.
  2. For y≠0 the formal quotient is x cot u/(3y).
  3. Over the reals, the logarithm requires 0<sin u≤1 and the square root requires ln(sin u)≥0. Thus sin u=1 and y=0 only at isolated inputs. The printed derivative has no real interval of validity.

📋 CBSE Stepwise Marking Scheme

  • [0.5 marks] Write u=x²/3+1 and formally differentiate y²=ln(sin u) to get 2yy′=(2x/3)cot u.
  • [2 marks] For y≠0 the formal quotient is x cot u/(3y).
  • [0.5 marks] Over the reals, the logarithm requires 0<sin u≤1 and the square root requires ln(sin u)≥0. Thus sin u=1 and y=0 only at isolated inputs. The printed derivative has no real interval of validity.

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All Questions in CBSE Class 12 Mathematics 2026 Supplementary Paper 65/7/3

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