



So we find the
derivative of some function. 


Say you have
... 


F(X)
= 3X^{5}  2X^{3} + X  6






The derivative
of this would be: 





F'(X)
= 15X^{4}  6X^{2} + 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 
d^{2}Y 

dX^{2} 






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)
= 3X^{5}  2X^{3} + X  6






F'(X)
= 15X^{4}  6X^{2} + 1






F''(X)
= 60X^{3}  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)
= 3X^{5}  2X^{3} + X  6 
original
function 
F'(X)
= 15X^{4}  6X^{2} + 1 
first
derivative 
F''(X)
= 60X^{3}  12X 
second
derivative 
F'''(x)
= 180X^{2}  12 
third
derivative 
F^{4}(X)
= 360X 
fourth
derivative 
F^{5}(X)
= 360 
fifth
derivative 
F^{6}(X)
= 0 
sixth
derivative 
F^{7}(X)
= 0 
seventh
derivative 
F^{8}(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
Y^{5} the fifth derivative of Y or Y to the 5th power? 





copyright 2005 Bruce
Kirkpatrick 
