Geometrically, the derivative is the slope of a curve. Physically, it represents the rate of change. The rate of change can be with respect to anything.
Details
The derivative is quite easy to understand really, as will be explained in the following example:
A vehicle, at time t = 0 seconds, is cruising along at 5 m/s (or about 11 mph). At 5 seconds, the car is moving along at 6 m/s. And so on, as shown in the following picture. At 20 seconds, the car is moving along at 21 m/s (or about 47 mph).
The following graph plots the vehicles speed versus time, as shown.
Simple enough. But now, one wants to find the rate of change of the vehicle's speed. Well first, what is rate of change? Rate of change is simply the rate at which something is changing, typically, vertical-axis values with respect to horizontal-axis values, or in this case, how the velocity is changing with respect to time. This is the rate of change. The words stating "velocity changing with respect to time" mathematically means velocity over time, or (m/s) / s, or m/s2, which is acceleration. So the rate of change of velocity with respect to time is acceleration. More about this can be found here.
As the speed of the vehicle is changing exponentially, the acceleration is probably changing over time as well (not constant over time). Therefore, the acceleration will have different values at different points in time like the velocity. An instantaneous acceleration can be found at any point crudely by approximating. This is done by taking one point just to the right of the value in question and another point just to the left of the value in question. In other words, lets look at time t = 2.5 seconds. The acceleration at 2.5 seconds can be crudely determined by finding the change in velocity from 5 m/s to 6 m/s divided by the change in time during that interval. Or:
f ' (2.5) =
6 m/s – 5 m/s
5 sec – 0 sec
= 0.2 m/s2
where f ' (2.5) is the derivative of the function f (t) at t = 2.5 seconds. The acceleration at 2.5 seconds is roughly 0.2 m/s2. This can be seen in the following graph, which shows the acceleration with respect to time at 2.5 seconds, as well as the accelerations approximated at 7.5 seconds, 12.5 seconds, and 17.5 seconds:
It just so happens that the velocity values in the first graph can be modeled by an equation. That equation is:
y(t) = 5 + 0.04t2
y(t), or just y, is the velocity and t is the time. There are relations that can be used to find the derivative, as a function, of another function, so that every single derivative value does not have to be calculated individually at every point desired. An equation can be used to model the derivative at any point. In this particular case the derivative of the previous equation is:
y ' (t) = 0.08t
Therefore, the acceleration, which is y ' (t), can be found by simply plugging in any t value into the equation instead of approximating.
So, by plugging in 2.5 seconds for t, it is found that the acceleration, y ' (t) = 0.2 m/s2, which is the same as the approximated value above. Note that this may not always be the case (the exact value equaling the approximated value, that is).
Here is another example which shows the function y = x2, where x can be distance, in meters.
The derivative of y = x2 is:
y ' = 2x
If the previous equation was distance with respect to time then this equation would represent velocity with respect to time. This equation is shown in the following graph:
If the derivative is taken again then the following equation results:
y '' = 2
Which is called the second derivative and would then be an equation for the acceleration, and actually the equation implies that the acceleration is constant. This equation is shown in the following graph:
A single asterisk, ', can signify a first derivative, or taking the derivative of a function once. A double asterisk, '', can signify a double derivative, or taking the derivative of a function once, then taking the derivative of that function. A derivative is both a noun and a verb, as a function can be a derivative, and if it is, it was obtained by taking the derivative of another function.
There are numerous relations that are used for taking the derivative of a function, depending on the structure of a function, and they will not be discussed here in this topic, as this serves as a basic introduction to derivatives.
Above, in the example with the car, an average value was taken to approximate the derivative, but technically, the derivative of a function at a point (a) is represented as:
f ' (a) =
lim
h→0
f(a + h) – f(a)
h
This is the general equation for the rate of change of f at a. There is more than one way to express the derivative, in terms of nomenclature. The first way, that was used in this lesson is:
(the derivative of y) = y '
another way is:
(the derivative of y) =
dy
dx
which means the change in y divided by the change in x, which is what was discussed in this lesson. This form is useful for equation manipulation, where for example if:
dy
dx
= 1
then:
dy = dx
Related ▪ L - Rectilinear Velocity
▪ L - Rectilinear Acceleration
