NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 3

✅ Step-by-Step Solution

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