Derivatives are a fundamental tool of calculus

Use the chain rule: $(f(g(x)))' = f'(g(x)) \cdot g'(x)$ .

Note that $\log^2 x$ is just a shorthand way to write $(\log (x))^2$ , so

$f(g)=g^2$ and $g(x)=\log(x)$

with derivatives

$f'(g) =2g$ and $g'(x)=\frac{1}{x}$ .

Putting the pieces together again, according to the chain rule, we get

$f'(g(x)) \cdot g'(x) = 2g(x) \cdot g'(x) = 2 \cdot \log(x) \cdot \frac{1}{x} = \frac{2\log(x)}{x}$