The problem asks for the derivative of the function $f(x) = x^2 \ln x$. We need to find $f'(x)$.

1. Problem Description

The problem asks for the derivative of the function $f(x) = x^2 \ln x$. We need to find $f'(x)$.

2. Solution Steps

We will use the product rule to find the derivative. The product rule states that if $f(x) = u(x)v(x)$, then $f'(x) = u'(x)v(x) + u(x)v'(x)$.

Let $u(x) = x^2$ and $v(x) = \ln x$.

Then $u'(x) = 2x$ and $v'(x) = \frac{1}{x}$.

Applying the product rule, we have:

$f'(x) = u'(x)v(x) + u(x)v'(x) = (2x)(\ln x) + (x^2)\left(\frac{1}{x}\right) = 2x \ln x + x$.

We can rewrite this as $x + 2x \ln x$.

3. Final Answer

The derivative of $x^2 \ln x$ is $x + 2x \ln x$.

The correct answer is D. $x + 2x \ln x$.