Class 12 Maths · Continuity And Differentiability · Long Answer #76

📋 CBSE Stepwise Marking Scheme

• [0.5 marks] For u=(sin x)^x, logarithmic differentiation gives u′=u[ln(sin x)+x cot x]. For v=arcsin√x, use v′=(1/√(1−x))·(1/(2√x)).
• [1 mark] Applying the indicated rules gives y′=\(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)} + \frac{1}{2 \sqrt{x} \sqrt{1 - x}}\).
• [1 mark] Collecting factors gives y′=\(\frac{2 x^{\frac{3}{2}} \sqrt{1 - x} \sin^{x}{\left(x \right)} \cos{\left(x \right)} + 2 \sqrt{x} \sqrt{1 - x} \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} \sin^{x}{\left(x \right)} + \sin{\left(x \right)}}{2 \sqrt{x} \sqrt{1 - x} \sin{\left(x \right)}}\).
• [0.5 marks] Therefore dy/dx=\(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)} + \frac{1}{2 \sqrt{x} \sqrt{1 - x}}\).

Master Chapter 5: Continuity And Differentiability