This picture shows us that the first point is known as x and the first y value for that point is known as f(x) because that is what the value of x is represented by. Then you have a second point that is h distance from x so you name it x+h (the first point plus the distance) and the y value for that one would then be f(x+h) [that is the x values name and since that means value of x you use whatever name we gave it]. Now we have two different points on a graph. With this we can plug our values into the slope formula which is y2-y1/x2-x1. This means if is f(x+h)-f(x)/x+h-x. The xs on the bottom cancel out and you are left with just an h on the denominator. Woah. Look at that. We have a formula that is known as the difference quotient.
On our graph we see that there is also a secant line. The thing is, we want a tangent line so we can use this formula to find the derivatives and specific slopes of that graph. The tangent line only touches the graph once. So the way that we get from secant to tangent is by setting the limit of h as 0. As we approach zero on the graph we get very very very very very very very very very close to the first point that we placed. The two points should overlap.
In this graph we see that the secant line has turned into a tangent line due to the limit that we gave it (I know these are two different graphs, but the concept remains the same for every graph that you have). The tangent line is then what we use for the rest of the concept. You use it to find the derivative and then the slope formula for the tangent line and then you can find the horizontal tangent and it is just so great. It is not too difficult though. Here is a video to assist you.
(https://www.youtube.com/watch?v=1mlkc3Pfxu4)
References
http://catalog.flatworldknowledge.com/bookhub/reader/4372?e=fwk-redden-ch04_s05
http://www.mathcaptain.com/calculus/difference-quotient.html
https://www.youtube.com/watch?v=1mlkc3Pfxu4
