Let a function $f(x)=y$ then for a small change $\delta x$ in x let there be a small change in y $\delta y$ i.e $f(x+\delta x)=y+\delta y$.
Then the incremental ratio will be $\delta x\over \delta y$ and $\lim_{\delta x\to 0}$=$\delta x\over \delta y$ will be the derivative of the function.(right?)
Now my question is: What does the incremental value tell us? For instance, when I divide it and get a numerical value, what does that numerical value tell me?
Suppose that, for example, $9\over3$ is the incremental value- What does that tell me?
Same goes with derivative, what its answer is telling me?
Asserting that the derivative at a point $x_0$ is $3$ means that for a small change of $x$ near $x_0$, $f(x)$ changes $3$ times more than that.
A more physical interpretation is this: if $f(t)$ is the distance (in meters) that an object is away from a fixed point after $t$ seconds, then $f'(t_0)=3$ means that the its speed at the moment $t_0$ is $3$ meters per second.
And, yes, the incremental ratio at $t_0$ is $f'(t_0)$.