How to use limit-derivative relationship for this specific case?

Question

Knowing that $f$ is such that $\ f(x + y) = f(x) + f(y) + xy,$

$lim_{h\to 0} \frac{f(h)}{h} = 2.$

How can we find the value of $f'(1)$ using informations that is given.

Answer 1

From your first equality

$$\frac{f(x+y)-f(x)}{y}=\frac{f(y)}{y}+x$$

Take the limit for $y\to 0$.