NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 11

✅ Step-by-Step Solution

1. Work where x cos x>0, x sin x>0 and x≠0.
2. For a positive base u, write uᵛ=exp(v log u). Its derivative is uᵛ(v′ log u+vu′/u). Here \(\frac{d}{d x} \left(x \cos{\left(x \right)}\right)^{x}\) = \(\left(x \cos{\left(x \right)}\right)^{x} \left(\frac{- x \sin{\left(x \right)} + \cos{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(x \cos{\left(x \right)} \right)}\right)\).
3. For a positive base u, write uᵛ=exp(v log u). Its derivative is uᵛ(v′ log u+vu′/u). Here \(\frac{d}{d x} \left(x \sin{\left(x \right)}\right)^{\frac{1}{x}}\) = \(\left(x \sin{\left(x \right)}\right)^{\frac{1}{x}} \left(\frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{x^{2} \sin{\left(x \right)}} - \frac{\log{\left(x \sin{\left(x \right)} \right)}}{x^{2}}\right)\).
4. Sum rule: \(\frac{d}{d x} \left(\left(x \sin{\left(x \right)}\right)^{\frac{1}{x}} + \left(x \cos{\left(x \right)}\right)^{x}\right)\) = \(\left(x \sin{\left(x \right)}\right)^{\frac{1}{x}} \left(\frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{x^{2} \sin{\left(x \right)}} - \frac{\log{\left(x \sin{\left(x \right)} \right)}}{x^{2}}\right) + \left(x \cos{\left(x \right)}\right)^{x} \left(\frac{- x \sin{\left(x \right)} + \cos{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(x \cos{\left(x \right)} \right)}\right)\).
5. Combining the terms gives y′ = \(\left(x \cos{\left(x \right)}\right)^{x} \left(\frac{- x \sin{\left(x \right)} + \cos{\left(x \right)}}{\cos{\left(x \right)}} + \log{\left(x \cos{\left(x \right)} \right)}\right) + \frac{\left(x \sin{\left(x \right)}\right)^{\frac{1}{x}} \left(\frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{\sin{\left(x \right)}} - \log{\left(x \sin{\left(x \right)} \right)}\right)}{x^{2}}\).