NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Miscellaneous Exercise Question 9

✅ Step-by-Step Solution

1. Work for π/4<x<3π/4.
2. For a positive base u, write uᵛ=exp(v log u). Its derivative is uᵛ(v′ log u+vu′/u). Here \(\frac{d}{d x} \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{\sin{\left(x \right)} - \cos{\left(x \right)}}\) = \(\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{\sin{\left(x \right)} - \cos{\left(x \right)}} \left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} - \cos{\left(x \right)} \right)} + \sin{\left(x \right)} + \cos{\left(x \right)}\right)\).
3. Combining the terms gives y′ = \(\left(\sin{\left(x \right)} - \cos{\left(x \right)}\right)^{\sin{\left(x \right)} - \cos{\left(x \right)}} \left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) \log{\left(\sin{\left(x \right)} - \cos{\left(x \right)} \right)} + \sin{\left(x \right)} + \cos{\left(x \right)}\right)\).