Derivatives Rates of Change

Average Rate of Change

Rate of Change is used to describe the ratio of a change in one variable compare to a change of another variable

If there is a function $f(x)=y$ then the rate of change of the function $f(x)$ correspond to the rate of change of variable z is

{\frac {\Delta y}{\Delta x}}={\frac {\Delta f(x)}{\Delta x}}

For example, if there are two points $(1,3)$ and $(2,7)$ , then

{\frac {\Delta y}{\Delta x}}={\frac {\Delta f(x)}{\Delta x}}={\frac {7-3}{2-1}}=4

The Rate of Change of a Function at a Point

Consider the formula for average velocity in the $x$ direction, ${\frac {\Delta x}{\Delta t}}$ , where $\Delta x$ is the change in $x$ over the time interval $\Delta t$ . This formula gives the average velocity over a period of time, but suppose we want to define the instantaneous velocity. To this end we look at the change in position as the change in time approaches 0. Mathematically this is written as: $\lim _{\Delta t\to 0}{\frac {\Delta x}{\Delta t}}$ , which we abbreviate by the symbol ${\frac {dx}{dt}}$ . (The idea of this notation is that the letter $d$ denotes change.) Compare the symbol $d$ with $\Delta$ . The idea is that both indicate a difference between two numbers, but $\Delta$ denotes a finite difference while $d$ denotes an infinitesimal difference. Please note that the symbols $dx$ and $dt$ have no rigorous meaning on their own, since $\lim _{\Delta t\to 0}\Delta t=0$ , and we can't divide by 0.

(Note that the letter $s$ is often used to denote distance, which would yield ${\frac {ds}{dt}}$ . The letter $d$ is often avoided in denoting distance due to the potential confusion resulting from the expression ${\frac {dd}{dt}}$ .)