An Introduction to Derivatives

Date: 05/17/98 at 14:15:43
From: Erika
Subject: Third derivative?

Hello, I have to research derivatives for a project. It asks about a 
third derivative. I haven't seen a second mentioned so I was 
wondering, is there a third derivative?

Thank you for your time.

Date: 05/20/98 at 22:44:11
From: Doctor Pat
Subject: Re: Third derivative?

Erika,

Yes, there is a second derivative, and a third, and a fourth... at 
least as long as the function continues to be differentiable. With 
polynomial functions you can take derivatives forever, but after long 
enough they all get kind of boring. I'll let you figure out why. 

A derivative is just the rate of change of one thing as measured by 
the change in another. 

Wow, I find that hard to understand and I wrote it. How about we look 
at a real world example?
  
If you go from here to there, you change your position. Let's pretend 
there is a number line along which you are walking, and a big clock on 
the wall. As you walk, your position and the clock are both changing.  
Your velocity (speed with a direction attached, but you knew that 
already, I suspect), which is the First derivative of position with 
respect to time, dP/dt, is how far you go divided by how long it 
takes. You may recognize this as the sixth grade formula for rate:

          distance
   rate = --------
            time

In calculus, we make the change in time smaller and smaller and use 
limits to come up with an "instantaneous" velocity. If you are moving 
with a constant speed the average speed (sixth grade method) and the 
dP/dt (really cool calculus method) are the same. 

So what is a second derivative? Well if you were walking along and 
decided to walk a little faster, the first derivative (your velocity) 
would also increase. The second derivative measures the change in the 
first derivative per unit of "whatever" (in our example, time).  

The common language for the change in velocity per unit of time is 
acceleration. When you step on the gas, you "change" your speed 
(velocity). How fast it is changing is the idea of a second 
derivative. And if your second derivative is changing, guess what we 
call that? We don't have names for all the different derivatives, or 
at least I don't know the names if they do exist, but you can see how 
each one is related to the one before it. Graphically you should also 
try to understand that if you graph y versus the nth derivative of x, 
then the slope of the curve at any point is the (n+1)th derivative of 
y with respect to x.  

Let's walk through the example with some computations of derivatives. 
If you do not know how to calculate derivatives and are not expected 
to, ignore all of this.  

If the position of some object is given as x(t) = t^3 + 2t + 1, then 
we can figure its position at any time by evaluating for the value 
of t.  At t = 0, the position is x = 1 at t = 1 the position is x = 4. 
We don't know how it got there, or where else it has been, but we know 
that between t = 0 and t = 1 it moved 3 units to the right. Its 
AVERAGE velocity was +3 units per time unit. If we take the derivative 
of x(t) with respect to time, we get the velocity. This is the FIRST 
derivative:

   x'(t) = 3t^2 + 2

We can use this function the same way we used the position function, 
only we get the velocity at each moment. For example at time t = 0 the 
velocity was 2, and at time t = 1 the velocity was 5. From this we can 
see that not only is it moving to the right, but it is speeding up.  
We can expect that it will move even farther to the right in the next 
unit of time than it did before.  

As before, now that we know two values of a function, we can figure 
the average acceleration (change in velocity/change in time) and get 
3 units per time unit squared. If we want the instanteous acceleration 
at any moment, we take the derivative of velocity, which is the second 
derivative of position:
 
   x''(t) = v'(t) = a(t) = 6t

This is a function which gives us the acceleration at any time. By now 
I think you get the picture. By the way, the third derivative of 
position with respect to time, which is the change in acceleration 
with respect to time, is named for what happens when you experience a 
sudden change of acceleration: you feel a jerk. 

Hope that helps. There are lots of ideas and lots of language involved 
in the ideas of calculus, but they really help to explain ideas that 
involve change and motion in a way earlier math could not. That is why 
calculus was so necessary for Kepler and Newton when they worked with 
the orbits of planets and the laws of gravity. 

If you need help on the mechanics of calculating a derivative, drop us 
another note. 

-Doctor Pat,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/   

Date: 05/22/98 at 08:19:42
From: Goodeffort
Subject: Re: Third derivative?

Thank you for helping me. The information you sent was just what I 
needed! :-)