Good morning. One example of a derivative looks like this. The velocity of an object in the
x direction equals the limit of the displacement of an object divided by change in time as the change
in time approaches zero. What?
Uh I agree.
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Right. That is usually how students respond. Let’s take a detailed look at what this means,
starting with this graph. The blue line represents the position as a function of time for an object.
Let’s talk about what happens to the object from time initial to time final. In other words,
during the change in time, delta t, represented on the graph. You can see the object starts
at the initial position, x i, and ends at the final position, x f, therefore, the object has
gone through a displacement, delta x. If I add a straight line in red to the graph which goes
from the initial to the final points, who can tell me what the slope of that line represents?
Well, that would be the change in position over the change in time for the object.
So that’s the velocity of the object. Actually, that is the average velocity
of the object during that change in time. Oh! That’s in the velocity derivative equation
you gave us before. That velocity equals the limit of average velocity as change
in time approaches zero, right mr.p? That is correct Billy. So far, we have
a red line on our graph which represents the average velocity of the object.
However, we need to discuss the limit of average velocity as change in time approaches zero.
In other words, let’s look at a specific point on the curve because at a specific point, the change
in time will be infinitesimally small and very close to zero. And we can add a green tangent line
to the blue motion curve at that specific point. This green tangent line represents the derivative
of this curve at this specific point in time and this specific location. So, what does the slope of
the green tangent line represent? Isn’t it just velocity?
Actually, it is instantaneous velocity. Oh, right. The slope of the red line is
the average velocity over that change in time and the slope of the green tangent line
is the instantaneous velocity at that specific instant in time.
Correct. Very nice.
The way we write the derivative for instantaneous velocity is dx over dt. What that means is
the derivative of x with respect to t or the derivative of x position with respect to time.
The definition of the derivative is the rate of change of a function with respect to a variable.
For our purposes, the derivative represents the slope of the tangent line
of a curve. And we just pointed out that slope of a position versus time graph
is instantaneous velocity, therefore the derivative of position with respect to time
equals instantaneous velocity. Please realize a derivative could also be the derivative of
velocity with respect to time or the derivative of linear momentum with respect to time,
or, well, there are a lot of options, which we will get into eventually. Be aware that usually
we are interested in the derivative with respect to time, however, that is not always the case
We could have the derivative of x position with respect to y position, for example. So, just
be careful to pay attention to what variable the derivative is being taken “with respect to”.
I do want to take a moment to discuss that the graph shows that the slopes of the lines
for the instantaneous velocity and average velocity currently have the same value.
Please realize this is only true at this one specific instant in time.
When we move the green tangent line for instantaneous velocity to the initial time for the average velocity,
you can see the instantaneous velocity at the initial time is larger than the average velocity over the change in time.
And, as we move forward in time, the slope of the tangent line decreases, and therefore the instantaneous velocity also decreases.
Only at this one specific point in time are the values of the instantaneous velocity and average velocity the same.
After this time, the instantaneous velocity is less than the average velocity.
Now let’s discuss how to actually take the derivative of a function. The derivative
rule we will start with is the derivative of a power function rule. It is the derivative
with respect to time of the function constant A times time to the power n equals n
times A times time to the power n minus 1. - What? Uh? Huh?
Yeah. Again, that is a typical response. Let’s do an example. If the function is
position as a function of time equals time squared, Bobby, please use the power rule
for derivatives to determine the derivative of that function with respect to time.
Uh well, the derivative of position with respect to time
equals the derivative with respect to time of time squared, in this case. Looking at the derivative
power rule, for this function, A equals nothing? A equals 1. 1 times time squared is still time
squared. So, the constant A in this function is 1. Right. A equals 1 and n equals 2.
So, the derivative equals n, which is 2, times A, which is 1, so it’s just 2,
times time to the power 2 minus 1. So, it equals 2 times time to the 1st power.
And time to the 1st power is just time. So, the derivative with respect to time of the function
position equals times squared equals 2 times time. I guess that makes sense.
But what does that mean? Thank you Bobby, you got the derivative
correct. And yes, let’s talk about what this means. Billy, remind me of what
the derivative of a function represents. The derivative of a function represents
the slope of the tangent line to a function. So, if the derivative of position equals time
squared is 2 times time ... Actually, no. Let’s graph those two functions.
Yeah. I think we can actually see how it works then. The slope of the function position equals
time squared at time equals zero is zero because the tangent line is completely horizontal there.
And two times time when time equals zero is zero. That’s why the derivative line starts out at zero.
Yeah. And the slope of a tangent line to position equals time squared increases
constantly from zero, which is exactly what the function 2 times time does.
Exactly. And we can look at two more specific points on our function.
At time equals 1, the slope of position equals time squared equals rise over run or 2 minus 0
over 1.5 minus 0.5, so the slope equals 2. And 2 times 1 also equals 2. So, at time equals 1,
the slope of the tangent line of our function equals the value of the derivative. And we can
do the same thing at time equals 2. The slope of position equals time squared equals rise over run
or 4 minus 0 over 2 minus 1 or 4 and 2 times 2 equals 4. So, at time equals 2, again,
the slope of the tangent line of our function equals the value of the derivative. Because
that is what the derivative represents. Yes Billy? Mr.p? It looks like, for dx dt or the derivative
of position with respect to time, position is on the y-axis and time is on the x-axis.
Is it always the case that the numerator is on the y-axis and the denominator is on the x-axis?
Good question Billy. What do y’all think? Well, the derivative represents the
slope of the tangent time. And slope equals rise over run.
Oh, so it does make sense. The rise is the numerator
and the run is the denominator. Cool. Thanks! Very nice. Alright, let’s do another example.
Bo, what is the derivative with respect to time of this function?
The function is y position as a function of time equals four times time to the third power
and we are determining the derivative of y position with respect to time.
Well, n is 3 and A is 4, so the derivative equals 3 times 4 times time to the power 3 minus 1 or 2.
So, the derivative equals 12 times time squared. Correct Bo. Billy, what about the derivative of
this x position as a function of y position equation?
Taking the derivative of x position with respect to y.
Okay, so x position as a function of y position equals 6 times y to the fourth power minus 5
times y plus 3. So, the derivative of x position with respect to y position equals
Well, 4 times 6 equals 24 times y to the 4 minus 1 equals 3, so, y to the third power. Minus
well, for 5y, A equals 5 but what does n equal? n equals 1. Y to the first power equals y.
Right. Thanks Bobby. You’re welcome.
So, the derivative with respect to y of 5y equals just 5.
But what is the derivative with respect to y of 3? I mean, y is not even in the term? Y is not in 3?
Yeah. Let’s talk about the derivative of a constant. For example, let’s look at the function
x equals 3. The graph of x position as a function of time looks like this. Looking at the graph,
what do you think the derivative of x with respect to time is when x equals 3?
Oh, the derivative represents the slope of the tangent line to a function.
And, the slope of the function equals zero because the line is horizontal the whole time.
So, the derivative of a constant equals zero because its slope is zero.
So, going back to our example, we add a “plus zero” to our derivative equation,
however, we do not need to because plus zero does not need to be there.
Mathematically, you could look at it as 3 times t to the zeroth power because t to the zeroth
power is 1 and n equals zero when you take the derivative of that with respect to time
and anything multiplied by zero equals zero. Very nice everybody. Another way to look at it is
that the derivative represents the rate of change of the function. When the function is constant,
it is not changing, so the rate of change of the function is zero. Okay, so class, change in
position over change in time equals …? Velocity.
Average velocity. Remember, that is average velocity.
Yeah, average velocity. And the derivative of position
with respect to time equals? Instantaneous velocity.
Right. So, change in position over change in time equals average velocity over a time period,
whereas the derivative of position with respect to time equals instantaneous velocity or the
velocity at a specific point in time. Now, class, change in velocity over change in time
equals …? average … acceleration.
Right… Average acceleration.
Therefore the derivative of velocity with respect to time equals …?
Instantaneous acceleration. Right. So, please notice this trend.
The change in a variable over change in time is the average value of that variable
and the derivative of a variable with respect to time is the instantaneous value of that variable.
We are going to see this many times in this class. And, just so you know, the derivative
of acceleration with respect to time is called the Jerk. The derivative of the jerk with respect to
time is called the Snap. The derivative of the snap with respect to time is called the Crackle.
Anybody want to guess what the derivative of the crackle with respect to time is called?
The pop. correct
How did you know?
Snap, crackle, and ... Pop.
Thank you very much for learning with me today, I enjoyed learning with you.