Question

Find the second order derivative of the function.

x³ log x

Answer 1

Let, y = x³ log x

Differentiating both sides with respect to x,

`dy/dx = x^3 d/dx log x + log x d/dx x^3`

= `x^3 * 1/x + log x * 3x^2`

= x² + 3x² log x

Differentiating both sides again with respect to x,

`(d^2 y)/dx^2 = d/dx x^2 + 3 [x^2 d/dx log x + log x d/dx x^2]`

= `2x + 3 [x^2 * 1/x + log x * 2x]`

= 2x + 3x + 6x log x

= 5x + 6x log x

= x (5 + 6 log x)