NCERT Class 12 Maths · Chapter 5 Continuity And Differentiability · Exercise 5.7 Question 5

✅ Step-by-Step Solution

1. Work for x>0.
2. Product rule: differentiate each variable factor once, keeping the other factors unchanged. Here \(\frac{d}{d x} x^{3} \log{\left(x \right)}\) = \(3 x^{2} \log{\left(x \right)} + x^{2}\).
3. Differentiate the first derivative \(3 x^{2} \log{\left(x \right)} + x^{2}\) again.
4. Power and chain rules: \(\frac{d}{d x} x^{2}\) = \(2 x\) × \(1\) = \(2 x\).
5. Product rule: differentiate each variable factor once, keeping the other factors unchanged. Here \(\frac{d}{d x} 3 x^{2} \log{\left(x \right)}\) = \(6 x \log{\left(x \right)} + 3 x\).
6. Sum rule: \(\frac{d}{d x} \left(3 x^{2} \log{\left(x \right)} + x^{2}\right)\) = \(6 x \log{\left(x \right)} + 5 x\).
7. Combining the terms gives y″ = \(x \left(6 \log{\left(x \right)} + 5\right)\).