Second Derivative of composite function

Question

I have $f'(x) = g(x^2)$ and $g'(x) = f(x^2)$ and must find $f''(x)$.

I believe if I take the second derivative of $f'(x) = g(x^2)$ then I have $f''(x) = g'(x^2) \cdot 2x$ using the chain rule. Or in terms of $f(x)$, $2x\cdot f(x^4)$.

Is this correct?

Thanks

Answer 1

Yes it’s correct

$$f'(x) = g(x^2) \implies f''(x) = g'(x^2) \cdot 2x \implies f’’(x)= 2x\cdot f(x^4)$$