Product Rule for Derivatives

Let $f$ be a three-times differentiable real function such that $f(x^2)\cdot f''(x) = f'(x)\cdot f'(x^2)$ for all real $x$. Given that $f(1) = 1$ and $f'''(1) = 8$, find the value of $f'(1) + f''(1)$.

Source: HMMT February 2007