Calculus Higher Derivatives
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Derivative Deja Vu
Higher Derivatives

 
So we find the derivative of some function.
Say you have ...

 F(X) = 3X5 - 2X3 + X - 6

 
The derivative of this would be:
 

 F'(X) = 15X4 - 6X2 + 1

  
 Could we find the derivative of this derivative?
 
 Why would you want to?
 
 Well who knows for now, but could it be done???
 
 Sure, why not
 Finding the derivative is just some dumb process you apply to a function.
 We've got a function.
 We've got a process.
 Let's do it!
 
 Wait a second...
 After you do find it,
 what are you going to call it?
 
 "The derivative of the derivative" is quite a mouth full.
 I'm sure those math types would have come up with a shorter name.
 
 And right you are!
 
The derivative of the derivative is called the second derivative.
 
The symbols we use for it look like this:
 
Y'' or F''(X) or d2Y

dX2
 
Take another look at where the exponents are on that dY and dX thing.
They are in different places.
Real fussy math types will get all over you if you slip up on that one. 
 
So:

 F(X) = 3X5 - 2X3 + X - 6

 

 F'(X) = 15X4 - 6X2 + 1

 

 F''(X) = 60X3 - 12X

 
 OK, so just what is a second derivative good for???
 
 Well, the first derivative is the rate of change.
 When we are talking about things that are moving, it is the velocity or speed
 So the second derivative is the rate of change of the speed.
 and that's known as acceleration.
 
 Like how fast will that fast and furious turbo razz matazz go from 0 to 60.
 
 In Physics class, they will get all over this stuff.
 Particles and billiard balls and cows all flying around bouncing off of each other.
 
 You'll love it!
 
 So could we take the derivative of the derivative?
 
 Sure, derivatives are like that battery bunny. 
 They just keep going and going.
 But sooner or later, they usually get boring.
 
 Examples:
F(X) = 3X5 - 2X3 + X - 6 original function
F'(X) = 15X4 - 6X2 + 1 first derivative
F''(X) = 60X3 - 12X second derivative
F'''(x) = 180X2 - 12 third derivative
F4(X) = 360X fourth derivative
F5(X) = 360 fifth derivative
F6(X) = 0 sixth derivative
F7(X) = 0 seventh derivative
F8(X) = 0 eighth derivative

 

You probably noticed that after the third derivative 

we started using numbers instead of prime marks.
I mean, do you really want to deal with F''''''''(X)?
While you can use any of the derivative symbols you want,
using the Y' notation when you get into higher derivatives could cause problems.
 Like, is Y5 the fifth derivative of Y or Y to the 5th power?
 

   copyright 2005 Bruce Kirkpatrick

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