Class 12 Maths · Continuity And Differentiability · Long Answer #133

📋 CBSE Stepwise Marking Scheme

• [0.5 marks] For u=(ln x)^x, ln u=x ln(ln x), giving u′=u[ln(ln x)+1/ln x]. For v=x^(x cos x), differentiate ln v=x cos x ln x.
• [1 mark] Applying the indicated rules gives y′=\(x^{x \cos{\left(x \right)}} \left(\left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(x \right)} + \cos{\left(x \right)}\right) + \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) \log{\left(x \right)}^{x}\).
• [1 mark] Collecting factors gives y′=\(- \frac{x x^{x \cos{\left(x \right)}} \log{\left(x \right)}^{2} \sin{\left(x \right)} - x^{x \cos{\left(x \right)}} \log{\left(x \right)}^{2} \cos{\left(x \right)} - x^{x \cos{\left(x \right)}} \log{\left(x \right)} \cos{\left(x \right)} - \log{\left(x \right)} \log{\left(x \right)}^{x} \log{\left(\log{\left(x \right)} \right)} - \log{\left(x \right)}^{x}}{\log{\left(x \right)}}\).
• [0.5 marks] Therefore dy/dx=\(x^{x \cos{\left(x \right)}} \left(\left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(x \right)} + \cos{\left(x \right)}\right) + \left(\log{\left(\log{\left(x \right)} \right)} + \frac{1}{\log{\left(x \right)}}\right) \log{\left(x \right)}^{x}\).

Master Chapter 5: Continuity And Differentiability