# What is the second derivative of f(x) = x^2ln x ?

Jan 7, 2016

$3 + 2 \ln x$

#### Explanation:

Let's get the first derivative f'(x) first by applying the product rule:
$f \left(x\right) = {x}^{2} \ln x$
$f ' \left(x\right) = {x}^{2} \left(\frac{1}{x}\right) + \left(\ln x\right) \left(2 x\right)$
$f ' \left(x\right) = x + 2 x \ln x$

Get the second derivative f''(x) by differentiating the first derivative:
Differentiate term by term. Apply product rule on the second term
$f ' \left(x\right) = x + 2 x \ln x$
$f ' ' \left(x\right) = 1 + 2 x \left(\frac{1}{x}\right) + \left(\ln x\right) \left(2\right)$
$f ' ' \left(x\right) = 1 + 2 + 2 \ln x$
$f ' ' \left(x\right) = 3 + 2 \ln x$