**Solution.**Let h(x) = \ln(x^2), f(u) = \ln(u) and g(x) = x^2. We will apply that chain rule:

\begin{align*} h'(x) = f'(g(x))g'(x). \end{align*}

\begin{align*} f'(u) = \frac{1}{u} \quad \text{and} \quad g'(x) = 2x. \end{align*}

\begin{align*} h'(x) &= f'(g(x))g'(x) \\ &= \frac{1}{x^2} \cdot 2x \\ &= \frac{2}{x}. \end{align*}