Difference Quotient

What is the difference quotient?

The difference quotient of a function is $$ \frac{f(x+h)-f(x)}{h} $$

The quantity $ h $ represents the increment of the independent variable.

Understanding the difference quotient
A practical example

Understanding the difference quotient

Consider a function $ f(x) $ defined on an interval $ (a,b) $.

$$ f(x) $$

To better understand the meaning of the difference quotient, it helps to visualize the function on the Cartesian plane.

graph of the function on the Cartesian plane

Now choose a generic value $ h $ in the interval $ (a,b) $. This value represents a change, or increment, in the independent variable $ x $.

After increasing $ x $ by $ h $, the function takes the new value

$$ f(x+h) $$

The corresponding variation can also be represented graphically.

variation of the function f(x) on the Cartesian plane

The change in the dependent variable, that is, the change in the function value, is

$$ f(x+h)-f(x) $$

The ratio between the change in the dependent variable and the change in the independent variable is called the difference quotient.

$$ \frac{f(x+h)-f(x)}{h} $$

Geometrically, the difference quotient represents the slope of a secant line connecting two points on the graph of the function.

Note. The difference quotient is defined for every value of $ h $ except $ h=0 $. If $ h=0 $, the denominator becomes zero and the expression is undefined.

Given a function $ f(x) $ defined on an interval $[A,B]$, the difference quotient is equal to the slope of the secant line passing through the points A and B.

slope of the secant line

A practical example

Consider the function

\[ y=f(x)=x^2+2x \]

Now compute the difference quotient at the point with x-coordinate $ 2 $, using an increment $ h $.

Start with the definition of the difference quotient:

\[ \frac{\Delta y}{\Delta x}=\frac{f(2+h)-f(2)}{h} \]

First compute $ f(2+h) $ by substituting $ x=2+h $ into the function:

\[ f(2+h)=(2+h)^2+2(2+h) \]

Expand the expression:

\[ f(2+h) =(4+4h+h^2)+4+2h \]

\[ f(2+h) =8+6h+h^2 \]

Next compute $ f(2) $:

\[ f(2)=2^2+2 \cdot 2 \]

\[ f(2)=4+4=8 \]

Substitute the results into the difference quotient formula:

\[ \frac{f(2+h)-f(2)}{h} = \frac{(8+6h+h^2)-8}{h} \]

\[ \frac{f(2+h)-f(2)}{h} = \frac{6h+h^2}{h} \]

\[ \frac{f(2+h)-f(2)}{h} = \frac{h(6+h)}{h} \]

\[ \frac{f(2+h)-f(2)}{h} =6+h \]

This final expression represents the slope of the secant line to the parabola through the point with x-coordinate $ 2 $, as the increment $ h $ changes.

As $ h $ gets closer to zero, the secant line approaches the tangent line to the curve.