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.

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