NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 10

✅ Step-by-Step Solution

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