Question 109
The function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfies $f\left(x^{2}\right) f^{\prime\prime}(x) = f^{\prime}(x) f^{\prime}\left(x^{2}\right)$ for all real $x$. Given that $f(1) = 1$ and $f^{\prime\prime\prime}(1) = 8$, determine $f^{\prime}(1) + f^{\prime\prime}(1)$.