Introduction to the notion of derivative

In studying limit we observe what happens to the values of a function when the values of the independent variable become closer and closer to a certain value. If a function is defined for every value of its domain it is continuous there. Graphically it means that there is no hole, jump or infinite branch. Quantitatively the function has a value for every value of the independent variable that belongs to the domain of the function. In limit and continuity we have been observing some changes in the behavior of a function when the independent variable behaves in a certain way. A function might have a limit when the independent variable becomes closer and closer to a certain value. For other values of the independent variable the same function has no limit. The function is not continuous.

The notion of derivative allows us to study more systematically the notion of change in a function. It allows us to study the change at any value of a function. The slope of a function allows us to study the change in this function with respect to the change or the increase of the value of the independent variable. The slope of a line is the rate of change of the dependent variable with respect to the change of the independent variable. Since a non-linear function varies in different ways there is no precise method to define its slope. This leads to the study of the slope of a tangent line to a function.  In order to study the change of a function it is important to define the notion of rate of change or slope of a line. the slope of a secant line to a curve or average rate of change or speed and slope at a point of a curve or instantaneous rate of change.

Slope of a line

The notion of slope is familiar to the civil engineers when they build roads. They have to figure out what type of slope they have to give to a road especially when they build it on a hill or in mountains. They have to shape the road in the right slope because if the road is too steep the cars cannot climb it. The slope is calculated by taking the tangent of the angle opposed to the right angle in a right triangle where the hypotenuse is the side that is going to be inclined. The slope is the measure of the inclination.of the hypotenuse. Its measure is calculated by dividing the opposite side to the angle to the adjacent side :

The slope of a function is also called the rate of change of this function. The slope of a line is called the rate of change of this line. It is the rate of the increase of y to the increase of x. It is constant at any part of the graph. It can be positive, negative or equal to zero. The slope of a line is calculated by dividing the difference of the y-ordinates of two points of that line by the difference of the x-ordinates.

   Watch this video to get some understanding of the notion of slope:

     
Slope of a tangent line to a curve


We have a curve (C), a secant line (PQ) and a tangent line L to the curve at the point P. The problem is to find the slope of the tangent line at P. In order to do this we make the point Q become closer and closer to the point P. As the point Q becomes close to the point P the initial secant P occupies different positions. At each position the secant has a different slope, The slope of the tangent line is the limit of the slopes of the different positions of the secant (PQ). In order to come to this conclusion let's calculate the function that allows to find the slope of the secant line (PQ).

If you are interested in learning more about these concepts I recommend that you go to Center for Integral Development where you will find a course about Derivatives and different other courses.