ddx\frac{d}{dx} and d2dx2\frac{d^2}{dx^2}

Calculus Level 2

If ddxf(x)=g(x)\displaystyle\frac{d}{dx} f(x) = g(x) and ddxg(x)=f(x2)\displaystyle\frac{d}{dx}g(x) = f(x^2), then d2dx2f(x3)= ?\displaystyle\frac{d^2}{dx^2}f(x^3) =\ ?

(A)f(x6)(B)g(x3)(C)3x2g(x3)(D)9x4f(x6)+6xg(x3)(E)f(x6)+g(x3) \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

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