Relations between the solutions of a non homogenous second order difference equation and their derivative? Here's an excerpt of a lecture note I am reading (I've highlighted the beginning of the part that I don't understand):

I don't understand how derivative comes into the picture.
Here's some context on my situation:
My Algebra course has covered (very briefly) non-homogenous second order difference equation. I want to understand to intelligently make a guess when finding a particular solution instead of memorizing all the possible forms. Nowhere during my lecture was derivative mentioned so I am really confuse about this point. Also, I have not yet covered ordinary differential equations, so if the answer is dependent on knowing ordinary differential equations, please mention which ideas that it specifically making reference to. 
 A: The heuristics is to compare the behaviour of the functions $t\mapsto x(t)$ solving the differential equation $x'(t)=F(x(t))$ to the sequences $(x_n)$ solving the difference equation $x_{n+1}=x_n+F(x_n)$. If $x(0)=x_0$, one can hope that $x(t)$ at time $t=n$ stays close to $x_n$, at least for values of $n$ that are not too large. This hope is based on the observation that, by definition of the derivative $x'(t)$, $x(t+s)=x(t)+F(x(t))s+o(s)$ when $s\to0$, which suggests that $x(t+1)$ might be close to $x(t)+F(x(t))$. Naturally, these are only approximations, whose quality may worsen when $t$ or $n$ become large.
A general remark is that while the differential equation $x'(t)=F(x(t))$ is often exactly solvable using usual functions, the difference equations $x_{n+1}=G(x_n)$ very seldom are, hence, despite the warning above about accumulating errors, it is often a good strategy to study the former for $G:\xi\mapsto F(\xi)+\xi$ to get some information (even if only qualitative) about the latter for $F$.
