What is the instantaneous rate of change in the real world? I can't grasp this concept of an instantaneous change of rate. How could a point on a function graph have a rate of change in the first place?
In this moment I just know that it is named the derivative and that it is the slope of the tangent line at that point. We can find that slope by finding the limit of closer and closer to the point slopes. I still don't understand what does it really represent. 
 A: Consider this example: You get into your car and drive around. The function $f(t)$ tells you how much distance you have covered until time $t$. Now the derivative of this function is your momentaneous velocity. 
So, since you only asked for an interpretation, think about this: Did you ever sit in a car wondering how fast you were going 'in that instance'? Of course, technically speaking, the tachometer only gives you some average speed over the last second or so but one still often thinks of it as the speed of the car 'right now'.
A: The functions we are working with in calculus are (at least at the introductory level) continuous. This basically means that no matter how far in we zoom on them, they will still be smooth. This is why we can find the instantaneous rate of change at some point; we can always just look at some ever smaller interval $\Delta x$ and compute the value of the slope $\frac{\Delta y}{\Delta x}$. If the function is continuous, then we can do this for as small a $\Delta x$ as we want, so if we let $\Delta x \rightarrow 0$, we get the limit that the slope at that point tends toward and we call that $\frac{dy}{dx}$ or instantaneous rate of change.
In the real world, things are never really continuous, but often it is a fine approximation, so we can use the tools we learn from calculus anyways.
