NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.5 Question 2

✅ Step-by-Step Solution

1. Work where (x−1)(x−2)/[(x−3)(x−4)(x−5)]>0; the real-domain boundary zeros x=1,2 do not have a finite one-sided derivative.
2. Power and chain rules: \(\frac{d}{d x} \sqrt{\frac{\left(x - 2\right) \left(x - 1\right)}{\left(x - 5\right) \left(x - 4\right) \left(x - 3\right)}}\) = \(\frac{1}{2 \sqrt{\frac{\left(x - 2\right) \left(x - 1\right)}{\left(x - 5\right) \left(x - 4\right) \left(x - 3\right)}}}\) × \(\frac{x - 2}{\left(x - 5\right) \left(x - 4\right) \left(x - 3\right)} + \frac{x - 1}{\left(x - 5\right) \left(x - 4\right) \left(x - 3\right)} - \frac{\left(x - 2\right) \left(x - 1\right)}{\left(x - 5\right) \left(x - 4\right) \left(x - 3\right)^{2}} - \frac{\left(x - 2\right) \left(x - 1\right)}{\left(x - 5\right) \left(x - 4\right)^{2} \left(x - 3\right)} - \frac{\left(x - 2\right) \left(x - 1\right)}{\left(x - 5\right)^{2} \left(x - 4\right) \left(x - 3\right)}\) = \(\frac{\sqrt{\frac{\left(x - 2\right) \left(x - 1\right)}{\left(x - 5\right) \left(x - 4\right) \left(x - 3\right)}} \left(x - 5\right) \left(x - 4\right) \left(x - 3\right) \left(\frac{x - 2}{2 \left(x - 5\right) \left(x - 4\right) \left(x - 3\right)} + \frac{x - 1}{2 \left(x - 5\right) \left(x - 4\right) \left(x - 3\right)} - \frac{\left(x - 2\right) \left(x - 1\right)}{2 \left(x - 5\right) \left(x - 4\right) \left(x - 3\right)^{2}} - \frac{\left(x - 2\right) \left(x - 1\right)}{2 \left(x - 5\right) \left(x - 4\right)^{2} \left(x - 3\right)} - \frac{\left(x - 2\right) \left(x - 1\right)}{2 \left(x - 5\right)^{2} \left(x - 4\right) \left(x - 3\right)}\right)}{\left(x - 2\right) \left(x - 1\right)}\).
3. Combining the terms gives y′ = \(- \frac{\sqrt{\frac{\left(x - 2\right) \left(x - 1\right)}{x^{3} - 12 x^{2} + 47 x - 60}} \left(x^{4} - 6 x^{3} - 5 x^{2} + 72 x - 86\right)}{2 \left(x - 5\right) \left(x - 4\right) \left(x - 3\right) \left(x - 2\right) \left(x - 1\right)}\).