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

✅ Step-by-Step Solution

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