If $f$ is a function defined for all real $x$ and satisfies $f(x+y) = f(x) + f(y) + xy(x+y)$, do $f'$ and $f''$ exist? 
Let $f$ be a function defined for all real $x$ and satisfies $f(x+y) = f(x) + f(y) + xy(x+y)$. If $f'(0) =-1$, then prove that $f$ and $f'$ are differentiable for all $x$. Also find $f'(3)$ and $f'(4)$.

I have tried assuming $x=y=0$ and solving the equation but to no avail. I also tried differentiating both sides. I just can't get a thought on its solution. Please help!

Let $$g(x):=f(x)-\frac{x^3}{3}$$ for all $x\in\mathbb{R}$.  Then, $g$ satisfies Cauchy's functional equation $$g(x+y)=g(x)+g(y)$$ for all $x,y\in\mathbb{R}$ with the additional property that $g$ is continuous at $0$ (since $g'(0)=f'(0)=-1$).  Thus, there exists $k\in\mathbb{R}$ such that $$g(x)=kx$$ for all $x\in\mathbb{R}$.  Finally, as $k=g'(0)=-1$, we get $$f(x)=-x+\frac{x^3}{3}$$ for all $x\in\mathbb{R}$.
