Just look carefully! 2

A function $f$ is such that:

$\large \begin{cases} yf(x) + x^2f(y) = f(xy) \\ \displaystyle \lim_{x \to 0} \frac{f(x)}{x} = 1 \end{cases}$

Prove that $f$ is differentiable infinitely many times.

Submit your answer as the value of $f'(10) - 10 f''(20)$ .