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.
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 $F$.