$\frac{d}{dx}$ and $\frac{d^2}{dx^2}$

If $\displaystyle\frac{d}{dx} f(x) = g(x)$ and $\displaystyle\frac{d}{dx}g(x) = f(x^2)$ , then $\displaystyle\frac{d^2}{dx^2}f(x^3) =\ ?$

$\begin{aligned} (A) &\quad f\left(x^6\right) & (B)&\quad g\left(x^3\right)\\ (C) &\quad 3x^2 g\left(x^3\right) & (D) &\quad 9x^4 f\left(x^6\right)+6x g\left(x^3\right)\\ (E) &\quad f\left(x^6\right) + g\left(x^3\right) & & \end{aligned}$

Credit: 1969 AP Calculus AB Exam