This is a challenging differentiation problem that requires a thorough understanding of the chain rule and the properties of derivatives. To solve this, we can break it down into smaller steps.
First, we can use the chain rule to rewrite the given equations as:
\frac{\,df(x)}{\,dx} = g(x) = g(x^3) \cdot \frac{\,dx^3}{\,dx}
and
\frac{\,dg(x)}{\,dx} = f(x^2) = f(x^6) \cdot \frac{\,dx^6}{\,dx}
Next, we can apply the chain rule again to find the second derivative:
\frac{\,d^2f(x^3)}{\,dx^2} = \frac{\,d}{\,dx} (g(x^3) \cdot \frac{\,dx^3}{\,dx}) = \frac{\,dg(x^3)}{\,dx} \cdot \frac{\,dx^3}{\,dx} + g(x^3) \cdot \frac{\,d^2x^3}{\,dx^2}
Similarly, we can find the second derivative of g(x^3):
\frac{\,d^2g(x^3)}{\,dx^2} = \frac{\,d}{\,dx} (f(x^6) \cdot \frac{\,dx^6}{\,dx}) = \frac{\,df(x^6)}{\,dx} \cdot \frac{\,dx^6}{\,dx} + f(x^6) \cdot \frac{\,d^2x^6}{\,dx^2}
Substituting these values into our original equation, we get:
\frac{\,d^2f(x^3)}{\,dx^2} = (f(x^6) \cdot \frac{\,d^2x^6}{\,dx^2}) \cdot \frac{\,dx^3}{\,dx} + (g(x^3) \cdot \frac{\,d^2x^3}{\,dx^2}) \cdot \frac{\,dx^6}{\,dx}
Since we know that \frac{\,d^2x^n}{\,dx^2} = n(n-1)x^{n-2}, we
