Imagine you have put two points on the line of a graph. You might want to calculate the average slope between these two points, that is how steep a straight line drawn between these two points is.

The way you do this is you take how far the two points are away from each other in the y-axis and divide that by how far the two points are from each other in the x-axis. This is commonly written as Δy/Δx.

Now imagine you start to move one of the points closer to the other one. As they get closer the value of the slope is going to start approaching whatever the slope is at exactly the first point.The problem is that we cannot get the points right on top of each other because then the difference in x-axis between the two points would be zero and we'd end up trying to divide by zero. Instead we see what happens as we get the points closer and closer to each other and observe what value the slope approaches as we do.

This value is what we call the limit of the slope as the distance between the points approaches zero. It's what we call the derivative of the function in that point. However unlike Δy/Δx it's not really a proper fraction.It's what that fraction approaches as we make x infinitely small. And to mark that it's not actually a proper fraction but rather the limit of a fraction we denote it as dy/dx.