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 text(/) 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`.