### Differentiation: The derivative

### The difference quotient

Change plays a major role in the study of mathematics and in particular the study of functions. In the following example, we look at the average change at a chosen interval.

We calculate the average change on the interval #[2,4]# for the function #f(x)=2x-2#.

The horizontal change #\blue{\Delta x}# is: \[\blue{\Delta x} = 4 - 2 = \blue{2}\] The vertical change #\green{\Delta y}# is: \[\green{\Delta y} = f(4)-f(2)=6 - 2 = \green{4}\] So the average change is:

\[\frac{\green{\Delta y}}{\blue{\Delta x}} = \frac{\green{4}}{\blue{2}}=2\]

Note that the notation #[a,b]# for an interval can also be used for coordinates.

We also call the average change on an interval the difference quotient.

Difference quotient

The **difference quotient **of a function #f# on an interval #[a,b]# is given by:

\[\dfrac{\green{\Delta y}}{\blue{\Delta x}}=\dfrac{\green{f(b)-f(a)}}{\blue{b-a}}\]

#\begin{array}{rcl}\dfrac{\Delta y}{\Delta x}&=&\dfrac{f(6)-f(1)}{6-1}\\&&\phantom{xxx}\blue{a=1 \text{ and }b= 6}\\

&=&\dfrac{(2\cdot 6^3+3\cdot 6 + 5)-(2\cdot1^3+3\cdot1 + 5)}{6-1}\\&&\phantom{xxx}\blue{x=1 \text{ and } x=6 \text{ substituted in } f}\\ &=& \dfrac{445}{5}\\&&\phantom{xxx}\blue{\text{added}}\\ &=&89\\&&\phantom{xxx}\blue{\text{divided}}\end{array}#

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