Difference Quotient

In single-variable calculus, the difference quotient is:

(1) $\begin{equation*} \frac{f(x + \Delta x) - f(x)}{\Delta x} \end{equation*}$

This tells us the average rate of change of function $f$ over the interval $\Delta x$ .

If we take the limit of the difference quotient as $x$ approaches 0:

(2) $\begin{equation*} \lim_{\Delta x \to 0} \frac{f(x + \Delta x) - f(x)}{\Delta x} \end{equation*}$

we get the instantaneous rate of change of function $f$ , better known as the derivative of $f$ .