Introduction

Introduction to differentiation: Definition of differentiation

Introduction

For various kinds of problems it is useful to know the rate change of a function.

In physics, for example, you can calculate the velocity of an object using a graph of position as a function of time.
In economics there are many applications of these changes in order to calculate the marginal benefit and marginal costs.
Often you can use the slope of the graph of a function (which measure the rate of change) in order to calculate the extremes of that function.

In this chapter we will learn how to calculate the change of a function.

One way to capture the change of a function at a point, is the tangent line.

Tangent line

Consider a graph containing a point $P$ . A line $l$ through $P$ is a tangent line at $P$ to the graph if:

in a small neighborhood around $P$ , the point $P$ is the only point that the line $l$ has in common with the graph;
in a small neighborhood around $P$ , the points of the line $l$ all lie on the same side of the graph.

The graph is often, but not always, obtained from a function.

The tangent line at $P$ to a graph need not exist and need not be unique. If it is unique, then the slope of the tangent line $l$ is a good measure of the rate of change of the graph at the point $P$ and we can consider the slope of the graph at that point $P$ .

Below are shown the graph of a quadratic function $f$ and the tangent to $f$ at a point $\rv{p,f(p)}$ . Move the points $\rv{a,f(a)}$ and $\rv{b,f(b)}$ to see how the tangent line moves along.

The slope of the tangent line provides information on how quickly the value of the function $f(x)$ changes when we change the variable $x$ . When we fix a point $P$ of the graph and know the slope of the graph at that point, then the tangent line is determined. We will find out how we can calculate the slope of the tangent line at $P$ and therefore the slope of the function at $P$ .

Below you see the graph $f(x)=2x^2-x$ . Additionally, the lines $a$ , $b$ , $c$ , and $d$ are drawn.

Which of these lines is/are tangent lines of the graph $f(x)$ ?

Line $a$ and line $d$ are tangent lines to the graph.

Line $a$ is a tangent line at $x=-1.76$ : the line has only that point in common with the graph, and lies entirely below the graph around that point.

Line $d$ is a tangent line at $x={{1}\over{4}}$ : the line has only that point in common with the graph, and lies entirely below the graph around that point.

Line $b$ and line $c$ are not tangent lines, because they do not have a small interval in which there is only one point of intersection with the graph and in which all other points are on one side of the graph.

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