Computing an instantaneous rate of change of any function

We can get the instantaneous rate of change of any function, not just of position. If $f$ is a function of $x$, then the instantaneous rate of change at $x=a$ is the limit of the average rate of change over a short interval, as we make that interval smaller and smaller. In other words, we want to look at $$\lim_{x \to a} \frac{\Delta f}{\Delta x} = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}.$$ This is the slope of the line tangent to $y=f(x)$ at the point $(a,f(a))$. It can also be written as a limit $$\lim_{h \to 0} \frac{f(a+h)-f(a)}{h},$$ where $h$ is just another name for $x-a$.

If this limit exists, we call it the derivative of $f$ at $x=a$. In the following video, we use this definition to compute the instantaneous velocity at time $t=2$ of a particle with position $s(t)=4t^2+3$.