Find all functions $f: \mathbb R\rightarrow \mathbb R$, at the same time satisfying the following two conditions:
a) $f (x + yf (x)) = f (x) f (y)$
b) the function $f$ can be represented in the form $f (x) = (\varphi (x)) ^ 2, x \in \mathbb R,$ where the function $f$ has a finite derivative at $x = 0.$ (not infinite)
I have no clue how to start. Any kind of help will be appreciated.