Afin Functions and Derivative

The simplest functions are lineal functions. Their formulas are polynomials with degree one or cero (this is the case when the function is the constant function). Their graphs are straight lines.

Two points determine a stright line. As a function we call it a linear function. We can see the slope of a line and how we can get the equation of a line through two points. We study also the x-intercept and the y-intercept of a linear equation.

We are interested in studying the derivative of simple functions with an intuitive and visual approach. We start with the linear function.

THE CONCEPT OF DERIVATIVE OF A FUNCTION

The derivative of a function at a point can be defined as the instantaneous rate of change or as the slope of the tangent line to the graph of the function at this point. We can say that this slope of the tangent of a function at a point is the slope of the function.

The slope of a function will, in general, depend on x. Then, starting from a function we can get a new function, the derivative function of the original function.

The process of finding the derivative of a function is called differentiation.

The value of the derivative function for any value x is the slope of the original function at x.

How can we draw the derivative function of a given function (in our case, a linear function)?

The general procedure is simple: We start drawing the tangent line to the function at a given point.

In our case it is very simple because the tangent line to a straight line is the the same line:

Then we draw a parallel line to the tangent passing through the value x-1 and we get a right triangle. The length of the vertical side is the slope of the tangent.

Then we can draw the derivative function of a linear function than is very easy because it is a constant function. The value of this constant function is the slope of the original linear function.

For example, a linear function with positive slope:

Another example, a linear function with negative slope:

When the function is the constant function, it is to say, its graph is an horizontal line (the slope is 0). Then, the derivative of a constant function is the constant function 0.

One simple and interesting idea is that when we translate up and down the graph of a function (we add or subtract a number from the original function) the derivative does not change. The reason is very intuitive and we can play with the interactive application to see this property. When you move the violet dot you are translating up and down the graph and the derivative is the same:

It is important to notice that the derivative of a polynomial of degree 1 is a constant function (a polynomial of degree 0). When we derive such a polynomial function the result is a polynomial that has a degree 1 less than the original function.

When we study the integral of a polynomial of degree 1 we can see that in this case the new function is a polynomial of degree 2. One degree more than the original function.

These results are related to the Fundamental Theorem of Calculus.

REFERENCES

Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.

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