NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 8

✅ Step-by-Step Solution

1. Work for 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 \(\frac{d}{d x} \sin^{x}{\left(x \right)}\) = \(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)}\).
3. Differentiate the outer function and multiply by the inner derivative: \(\frac{d}{d x} \operatorname{asin}{\left(\sqrt{x} \right)}\) = \(\frac{1}{2 \sqrt{x} \sqrt{1 - x}}\).
4. Sum rule: \(\frac{d}{d x} \left(\sin^{x}{\left(x \right)} + \operatorname{asin}{\left(\sqrt{x} \right)}\right)\) = \(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)} + \frac{1}{2 \sqrt{x} \sqrt{1 - x}}\).
5. Combining the terms gives y′ = \(\left(\frac{x \cos{\left(x \right)}}{\sin{\left(x \right)}} + \log{\left(\sin{\left(x \right)} \right)}\right) \sin^{x}{\left(x \right)} + \frac{1}{2 \sqrt{x} \sqrt{1 - x}}\).