To understand the concept of derivative, you first need to know what a tangent line is.
We fixed a point P on the graph of a function f, and we chose one Q 
P. Doing Q approach P, it may happen that the line PQ tends to a limit position: a straight line t.
In this case, t is called the tangent line of f in Pas long as it is not vertical. So the straight line PQ is called the secant line to the graph of f in P.
We can see from the chart below that Q should approach P left and right, and in both cases the straight line PQ should tend to t (green straight).
First chart - To the left
Second graph - By the right
NOTE: The graph tangent line of a function does not always exist.
The figure below shows an example of a graph where P is the nozzle of a function, so the process described above leads to two limit positions (t1 and t2), obtained respectively by making Q approach P left and right.
Calculation of the slope of the tangent line
Consider the curve that is the graph of a continuous function. f and P (xo, f (xo)) a point on the curve. We will now analyze the calculation of the slope (angular coefficient) of the line tangent to the curve drawn by f on point P.
To look at this question, we chose a small number 
x, other than 0, where 
x is the displacement on the abscissa axis. On the chart we mark the point Q (xo + 
x, f (xo + 
x)). We draw a secant line that goes through the points P and Q.
The slope (angular coefficient) of this line is given as follows:
