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 **P**as 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:

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