NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Miscellaneous Exercise Question 11

✅ Step-by-Step Solution

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