NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 7

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