The difference quotient at a point

Differentiation: The derivative

The difference quotient at a point

Using the difference quotient, we can approach the change at one point of a graph.

We want to approach the change at point $\green{x}=\green{2}$ for the function $\blue{f(x)}=\blue{x^2}$ . To do so, we take the difference quotient on an interval around $\green{2}$ : $[\green{2},\green{2}+\orange{h}],$

in which we choose a smaller $\orange{h}$ each time.

The smaller $\orange{h}$ we choose, the better we can see the change at the point. We see that these values are getting closer and closer to $4$ as we choose smaller values of $\orange{h}$ . The change at a point is also called the slope.

We can approach the change at a point $x=a$ for each function by determining the difference quotient on the interval $[a,a+h]$ .

The difference quotient on an interval of length h

For a function $\blue{f}$ , the difference quotient at point $\green{x}=\green{a}$ with difference $\orange{h}$ is defined as follows:

$\dfrac{\Delta y}{\Delta x}= \dfrac{f(\green{a}+\orange{h})-f(\green{a})}{\orange{h}}$

We can leave $\orange{h}$ as is in our calculation.

Example

$\blue{f(x)}=\blue{x^2}$ and $\green{a}=\green{4}$ give:

$\begin{array}{rcl}\dfrac{\Delta y}{\Delta x}&=& \dfrac{(\green{4}+\orange{h})^2-\green{4}^2}{\orange{h}}\\&=& \dfrac{16+8\cdot\orange{h}+\orange{h}^2-16}{\orange{h}} \\&=& \dfrac{8\cdot\orange{h}+\orange{h}^2 }{\orange{h}} \\&=& 8+\orange{h}\end{array}$

Give the difference quotient at point $x=6$ with difference $h$ of the function $f(x)=3 x + 1$ . Simplify your answer as much as possible.

$\frac{\Delta y}{\Delta x}=$ $3$

$\begin{array}{rcl}\dfrac{\Delta y}{\Delta x}&=& \dfrac{f(6+h)-f(6)}{h}\\&&\phantom{xxx}\blue{\text{definition difference quotient at }x=6 }\\&=&\dfrac{3\cdot(6+h)+1-(3\cdot6+1)}{h}\\&&\phantom{xxx}\blue{6 \text{ entered in }f}\\ &=&\dfrac{18+3\cdot h -18}{h}\\&&\phantom{xxx}\blue{\text{expanded brackets}}\\&=&\dfrac{3\cdot h}{h}\\&&\phantom{xxx}\blue{\text{added}}\\&=&3\\&&\phantom{xxx}\blue{\text{eliminated }h}\end{array}$

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