## Precalculus (6th Edition) Blitzer

The instantaneous rate of change of a function is obtained by taking limit $h\to 0$ in the average rate of change of a function.
The average rate of change of a function from $x=a\text{ to }x=a+h$ is given by $\frac{f\left( a+h \right)-f\left( a \right)}{h}$. This is, the slope of the secant line between $\left( a,f\left( a \right) \right)\text{ and }\left( a+h,f\left( a+h \right) \right)$. The instantaneous rate of change of a function at a point $\left( a,f\left( a \right) \right)$ is given by $\underset{h\to 0}{\mathop{\lim }}\,\frac{f\left( a+h \right)-f\left( a \right)}{h}$. This is the slope of the tangent line at $\left( a,f\left( a \right) \right)$. Thus, the instantaneous rate of change of a function is obtained by taking limit $h\to 0$ in the average rate of change of a function.