Difference quotient

Introduction: Quite often we are interested in the change of the function value $y(x)$ when the variable $x$ changes. Such a change in $x$ is denoted by $\Delta x$. Hence, the variable has the new value
$$x + \Delta x.$$
If $\Delta x$ is positive, we speak of an increase; if $\Delta x$ is negative, we speak of a decrease. The new function value can be found by plugging $x+\Delta x$ into the function; the new function value is therefore
$$y(x+\Delta x).$$
We denote the change in the function value by
$$\Delta y = y(x+\Delta x) - y(x).$$
Then the average change of the function value for a change of $x$ by $\Delta x$ is the quotient of $\Delta y$ and $\Delta x$; we call this quotient the difference quotient.

Definition: The difference quotient of a function $y(x)$ in the point $x$ at a change $\Delta x$ is the quotient:
$$\dfrac{\Delta y}{\Delta x} = \dfrac{y(x+\Delta x) - y(x)}{\Delta x}.$$

We can show the difference quotient also in a figure, as shown below. Here the difference quotient of the function $y(x)$ is equal to the slope of the line through the points $P(x,y(x))$ and $Q(x+\Delta x,y(x+\Delta x))$.