Second derivative as bilinear form

Question

I have a question concerning the second derivative. Consider the ode $ y' = f(y)$. Then we have

$ y'' = f'(y) f(y)$

with the notation $f'(y) = f_y(y)$ beeing the derivative with respect to variable y.

Now it follows $ y''' = f''(f,f) + f'f'f. $

How ist $(f,f)$ exactly to be understood? Is the second derivative beeing considered as a bilinearform? If yes, in which book is it well explained?

Thanks in advance for every answer.