Differentiate $3x^2\ln x$.

1. Apply the product rule to differentiate the expression

Let $y = 3x^2\ln x$.
Using the product rule $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ with $u = 3x^2$ and $v = \ln x$:
$\frac{dy}{dx} = 3x^2 \cdot \frac{d}{dx}(\ln x) + \ln x \cdot \frac{d}{dx}(3x^2)$

2. Differentiate the individual terms and simplify

$\frac{dy}{dx} = 3x^2 \left(\frac{1}{x}\right) + (\ln x)(6x)$
$= 3x + 6x\ln x$