Tangent line

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:

Next: What is derived?