Find f(x) using definition of differentiability

Question

Let $f(x + y) = f(x) + f(y) + 5xy$ and $\lim_{h\to0}\frac {f(h)}{h}=3$. Find $f'(x)$.

I tried this:

We have $f(0)=0$. so then if $y\to-x=>f(0)=f(-x)+f(x)-5x^{2}$ so then:

$$f(x)+f(-x) = 5x^{2}$$.

But I get stuck here because this doens't really tell me too much... I need some hints. How would you recommend me to use the definition of differentiability?

Answer 1

$$f’(x)=\lim_{h \to 0} \frac {f(x+h)-f(x)}h \\= \lim_{h \to 0} \frac {f(x)+f(h)+5xh-f(x)}h \\= \lim_{h \to 0} \frac {f(h)+5xh}h $$