# Question #c67a6

Jun 10, 2015

If a mathematical equation describes some physical quantity as a function of time, the derivative of that equation describes the rate of change as a function of time.

#### Explanation:

For example, if the motion of a car can be described as:
$x = v t$
Then at any time ($t$) you can say what the position of the car will be ($x$). The derivative of $x$ with respect to time is:
$x ' = v$.
This $v$ is the rate of change of $x$.

This also applies to cases where the velocity is not constant. The motion of a projectile thrown straight up will be described by:
$x = {v}_{0} t - \frac{1}{2} g {t}^{2}$
The derivative will give you the velocity as a function of $t$.
$x ' = {v}_{0} - g t$
At time $t = 0$ the velocity is simply the initial velocity ${v}_{0}$. At later times, gravity will be constantly decreasing the velocity until it becomes zero and then negative.

But it's not limited to equations of motion. If you ask about decay rates of radioactive material, I can right down a function for the number of atoms at any given time:
$n = {n}_{0} {e}^{- \lambda t}$
And the rate at which I see atoms decay will be:
$n ' = - {n}_{0} \lambda {e}^{- \lambda t}$