The notion of difference quotient

Introduction to differentiation: Definition of differentiation

The notion of difference quotient

In order to calculate the slope of a tangent line to a graph, we can take a line that lies close to the tangent line. For this line, you can calculate the slope by using the difference quotient.

Below you see the graph of the function $f(x)=\frac{1}{2}x^2+5$ and the tangent line $l$ to $f$ at the point $\rv{2,7}$ . You can also see the line $m$ through $\rv{2, 7}$ and $\rv{3,{{19}\over{2}}}$ . Both points lie on the graph of $f$ .

Approximate the slope of the tangent line $l$ by calculating the slope of line $m$ .

The slope of the line $m$ through $\rv{2, 7}$ and $\rv{3, {{19}\over{2}}}$ is ${{5}\over{2}}$

The slope of line $m$ is given by:

$\begin{array}{rcl}\frac{\text{the difference in }y}{\text{the difference in }x}&=&\frac{\Delta y}{\Delta x}\\ &&\phantom{x}\color{blue}{\Delta y =\text{the difference between the }y\text{ coordinates}}\\ &&\phantom{x}\color{blue}{\text{of }\rv{3, {{19}\over{2}}}\text{ and }\rv{2, 7}}\\ &&\phantom{x}\color{blue}{\Delta x =\text{the difference between the }x\text{ coordinates}}\\ &&\phantom{x}\color{blue}{\text{of the same points}}\\ &=&\frac{{{19}\over{2}}-7}{3-2}\\ &=&{{5}\over{2}}\end{array}$
We can therefore approximate the slope of the tangent line $l$ by $3$ .

As we will see later in this chapter, the actual slope of $l$ is $2$ .

New example

Such an approximation of the slope of a tangent line of a graph is called a difference quotient. This is so called because it is actually the quotient of two differences $\Delta y$ and $\Delta x$ that express the change, $\Delta x$ in $x$ and the change, $\Delta y$ in $y$ , respectively.

Below the formal definition of the difference quotient is stated for the case where the graph is derived from a function.

The difference quotient

The difference quotient of a function $f$ at $a$ with difference $h$ is

$\frac{f (a+h)− f (a)}{h}\tiny.$

The difference quotient measures the growth of $f$ in the transition from $a$ to $a+h$ , normalized by the length $h$ of the interval.

The illustration below shows the difference quotient of $f$ at $a$ with difference $h$ .

As we see in the example and in the figure, the following rule applies:

Difference quotient and slope

The difference quotient of $f$ at $a$ with difference $h$ is equal to the slope of the line through the points $\rv{a+h,f(a+h)}$ and $\rv{a,f(a)}$ .

A motivating example is the linear function $f(x)=c\cdot x+d$ . The difference quotient of this function at $a$ with difference $h$ is

$\begin{array}{rcl}\frac{f(a+h)−f(a)}{h} &=&\frac{c\cdot(a+h)+d-(c\cdot a+d)}{h}\\&=&\frac{c\cdot a+c\cdot h+d-c\cdot a-d}{h}\\&=&\frac{c\cdot h}{h}\\&=&c\end{array}$ This is indeed the slope of the line $y=c\cdot x+d$ through $\rv{a,f(a)}$ and $\rv{a+h,f(a+h)}$ .