Section Summary

In this section we have seen that, for most functions \(f\), if you zoom in on the graph of $y=f\left(t\right)$ near a point $(t_0,y_0)$, the graph appears to be a straight line. This property of the function is called “local linearity”; the slope of the apparent straight line is the value of the derivative of \(f\) at $t_0$. From an algebraic point of view, the derivative of \(f\) at $t_0$ is the limiting value of the difference quotients \(\Delta y / \Delta t\) as $\Delta t$ approaches \(0\). Here $\Delta t$ is the difference $t_1-t_0$ between a nearby point $t_1$ and $t_0$, and $\Delta y$ is the corresponding difference of \(y\)-values.

We can think of the derivative as the instantaneous rate of change of \(f\) at $t_0$. In particular, if \(y\) represents distance and \(t\) represents time, the derivative is called “velocity.”

We denote the derivative by

if we want to emphasize the variable notation, or by

if we want to display the function name \(f\).