Class 12 Continuity And Differentiability · Long Answer3 Marksbranch_sensitive_differentiation
Class 12 Maths · Continuity And Differentiability · Long Answer #138
For y=arcsin[6x√(1−9x²)] with −1/(3√2)<x<1/(3√2), find y′.
✅ Step-by-Step Solution
Final Answer: 6/√(1−9x²)
- Let θ=arcsin(3x). The given interval puts θ in (−π/4,π/4), with cosθ=√(1−9x²).
- The argument becomes 2sinθ cosθ=sin2θ, and 2θ lies in the principal arcsine interval.
- Hence y=2θ=2arcsin(3x), so y′=2·3/√(1−9x²).
- Therefore y′=6/√(1−9x²).
📋 CBSE Stepwise Marking Scheme
- [0.5 marks] Let θ=arcsin(3x). The given interval puts θ in (−π/4,π/4), with cosθ=√(1−9x²).
- [1 mark] The argument becomes 2sinθ cosθ=sin2θ, and 2θ lies in the principal arcsine interval.
- [1 mark] Hence y=2θ=2arcsin(3x), so y′=2·3/√(1−9x²).
- [0.5 marks] Therefore y′=6/√(1−9x²).
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