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So we find the
derivative of some function. |
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Say you have
... |
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F(X)
= 3X5 - 2X3 + X - 6
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The derivative
of this would be: |
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F'(X)
= 15X4 - 6X2 + 1
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Could we
find the derivative of this derivative? |
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Why would
you want to? |
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Well who
knows for now, but could it be done??? |
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Sure, why
not |
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Finding
the derivative is just some dumb process you apply to a function. |
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We've got
a function. |
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We've got
a process. |
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Let's do
it! |
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Wait a
second... |
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After you
do find it, |
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what are
you going to call it? |
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"The
derivative of the derivative" is quite a mouth full. |
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I'm sure
those math types would have come up with a shorter name. |
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And right
you are! |
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The derivative
of the derivative is called the second derivative. |
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The symbols we
use for it look like this: |
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Take another
look at where the exponents are on that dY and dX thing. |
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They are in
different places. |
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Real fussy math
types will get all over you if you slip up on that one. |
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So: |
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F(X)
= 3X5 - 2X3 + X - 6
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F'(X)
= 15X4 - 6X2 + 1
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F''(X)
= 60X3 - 12X
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OK, so
just what is a second derivative good for??? |
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Well, the
first derivative is the rate of change. |
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When we
are talking about things that are moving, it is the velocity or
speed |
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So the
second derivative is the rate of change of the speed. |
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and
that's known as acceleration. |
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Like how fast will that fast and furious turbo razz matazz go from 0
to 60. |
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In
Physics class, they will get all over this stuff. |
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Particles
and billiard balls and cows all flying around bouncing off of each
other. |
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You'll
love it! |
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So could
we take the derivative of the derivative? |
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Sure,
derivatives are like that battery bunny. |
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They just
keep going and going. |
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But
sooner or later, they usually get boring. |
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Examples: |
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| 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 |
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You probably
noticed that after the third derivative |
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we started using numbers
instead of prime marks. |
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I mean, do you
really want to deal with F''''''''(X)? |
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While you can use any of the derivative symbols you
want, |
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using the Y'
notation when you get into higher derivatives could cause problems. |
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Like, is
Y5 the fifth derivative of Y or Y to the 5th power? |
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copyright 2005 Bruce
Kirkpatrick |
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