Class 12 Maths · Continuity And Differentiability · Long Answer #71

📋 CBSE Stepwise Marking Scheme

• [0.5 marks] For u=(sin x)^x use ln u=x ln(sin x); for v=(cos x)^(sin x) use ln v=sin x ln(cos x). Differentiate each logarithmic equation and add u′+v′.
• [1 mark] Applying the indicated rules gives y′=\(\left(\log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}}\right) \cos^{\sin{\left(x \right)}}{\left(x \right)} + \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)}\).
• [1 mark] Collecting factors gives y′=\(\frac{x \sin^{x}{\left(x \right)} \cos^{2}{\left(x \right)} + \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} \sin^{x}{\left(x \right)} \cos{\left(x \right)} + \log{\left(\cos{\left(x \right)} \right)} \sin{\left(x \right)} \cos^{2}{\left(x \right)} \cos^{\sin{\left(x \right)}}{\left(x \right)} - \sin^{3}{\left(x \right)} \cos^{\sin{\left(x \right)}}{\left(x \right)}}{\sin{\left(x \right)} \cos{\left(x \right)}}\).
• [0.5 marks] Therefore dy/dx=\(\left(\log{\left(\cos{\left(x \right)} \right)} \cos{\left(x \right)} - \frac{\sin^{2}{\left(x \right)}}{\cos{\left(x \right)}}\right) \cos^{\sin{\left(x \right)}}{\left(x \right)} + \left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)}\).

Master Chapter 5: Continuity And Differentiability