Class 12 Maths · Continuity And Differentiability · Long Answer #130

📋 CBSE Stepwise Marking Scheme

• [0.5 marks] For u=x^(cos x), ln u=cos x ln x. For v=(sin x)^(tan x), ln v=tan x ln(sin x). Differentiate these products, multiply by u,v respectively, and add.
• [1 mark] Applying the indicated rules gives y′=\(x^{\cos{\left(x \right)}} \left(- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) + \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sin{\left(x \right)} \right)} + \frac{\cos{\left(x \right)} \tan{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin^{\tan{\left(x \right)}}{\left(x \right)}\).
• [1 mark] Collecting factors gives y′=\(\frac{- x x^{\cos{\left(x \right)}} \log{\left(x \right)} \sin^{2}{\left(x \right)} + x \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} \sin^{\tan{\left(x \right)}}{\left(x \right)} \tan^{2}{\left(x \right)} + x \log{\left(\sin{\left(x \right)} \right)} \sin{\left(x \right)} \sin^{\tan{\left(x \right)}}{\left(x \right)} + x \sin^{\tan{\left(x \right)}}{\left(x \right)} \cos{\left(x \right)} \tan{\left(x \right)} + x^{\cos{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}}{x \sin{\left(x \right)}}\).
• [0.5 marks] Therefore dy/dx=\(x^{\cos{\left(x \right)}} \left(- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x}\right) + \left(\left(\tan^{2}{\left(x \right)} + 1\right) \log{\left(\sin{\left(x \right)} \right)} + \frac{\cos{\left(x \right)} \tan{\left(x \right)}}{\sin{\left(x \right)}}\right) \sin^{\tan{\left(x \right)}}{\left(x \right)}\).

Master Chapter 5: Continuity And Differentiability