



If we have the graph of a
straight line, like maybe Y = 2X 2, it looks like this:












The
RATE that the line goes up or down when looking from left to right 


is called the "SLOPE." 


Sometimes we
use the letter "m" as the symbol for the slope. 


Math types say:






Slope
of the line (m) = 
Change
in Y 
= 
DY 


Change
in X 
DX 






If you know two
points on the line, 


like maybe (0,2) and (1,0)



you can find
the slope of the line using the equation: 





Slope
= 
Y_{2}
 Y_{1} 
= 
0
 (2) 
= 
2 
=
2 



X_{2}
 X_{1} 
1
 0 
1 






You can chose either point as
point 1 (X_{1},Y_{1}) and you get the right answer.






Lots of the
examples in these pages use F(x) instead of Y. 


The slope
equation can use them too. 


That would look
like this: 


Slope
= 
F(x_{2})
 F(x_{1}) 

X_{2}
 X_{1} 






Since the graph of the example we
used above is a straight line,



you get the same slope no matter what two
points you chose.






BUT ...



If we have an equation where the
graph is not a straight line,



the story isn't so simple.






A curved line
doesn't have just one slope value for it's whole length. 


The slope
changes as the line curves. 





The slope of a curved line at any
point on the line



is the same as the slope of the straight line that
just touches,



but does not cross, the line at that point.






In math talk,
we say the line is "tangent" to the curve at that point. 





It looks like
this: 








OK, so why
would somebody WANT to find the slope of the line at some point? 





The slope is
the rate of change of the function at that point. 


Rates of change are used all
through science, statistics, economics,



finance, accounting, biology and
lots of other places.






We use rates of
change all the time. 


The reading on
a speedometer, the interest rate on a savings account or loan, 


the
inflation rate, the rate that germs are killed by some new medicine, 


there are zillions of
'em. 





No matter what
you do, you will be way ahead of most people 


if you really
understand how the rates of change operate in your subject. 





Oh, one more
bit of interesting trivia. 


The rate of
change at some point is called the derivative, 


and finding that value for a
point is called Differential Calculus.



Yeah, it's just
like those math types. 


Big scary name for something
that's no big deal.






Let's do an
example: 





Say we have the
function F(X) = X
^{2} (also known as Y = X ^{2}) 





OK, so right
off you probably notice that the slope is changing all the time. 


How can we calculate what the
slope is at any one point?






We could pick
two points that were really close together 


and find the slope of the
line between them. 


That would be
really close to the slope of the line near those points. 


But it probably
wouldn't be the exact slope of the line at the point we want. 





So what do we
do? 


Actually, we do
something very close to that. 





First, chose
the X value you want the slope for, 


then find the
function value at that point. 





Now, move a
little ways to the right of the point and chose another X value. 





Call the
distance between the two X values "the change in X." 


Math types use
a Greek letter delta ( D
) for
"change in." 


That means
"the change in X" is written as: DX 





So to tell our
two X points apart, we call one X and the other X + DX. 





That means our
two function values can be called F(x) and F(x + Dx). 


Putting this
all together in one graph, we get: 








So the two (X,Y)
type points we have made are: 








Math types are
always writing stuff like F(x + Dx)
to try to look so smart. 


Don't fall for
that one. 


Just say:
"Ha Ha!, you math types can't fool me with that one! 


That's just
Y_{2} in disguise!" 








Take the old
"using two points to find the slope" equation: 





Slope
= 
Y_{2}
 Y_{1} 

X_{2}
 X_{1} 






And use our
names for the values in the graph above and you get: 





Slope
= 
F(x +
Dx)
 F(x)


(x +
Dx)
 x







Simplifying the
denominator a bit: 





Slope
= 
F(x +
Dx)
 F(x)


Dx 






So we want to
find the slope of the line at the point X. 


We sneak up on
this by thinking of the two X values, 


X and X + DX,
as being really close together. 





The closer the
two points are to each other, 


the smaller the distance between them. 


(is that
a major DUH or what?) 





Anyway, that
distance is DX,
and we want to keep making it smaller and smaller. 





Now the two
points are getting closer and closer and DX
is getting smaller, 


and smaller. 


The point we
want is just as DX
gets to zero. 





But when DX
gets to zero, we have a big problem. 





In the
simplified slope equation we wrote above, the denominator is DX. 


(Scroll up and
check if you want) 





So we need to
sneak up on a zero in the denominator and try to get an answer 


for a
point that's really undefined. 





WAIT A MINUTE! 





That's what all that silly
LIMITS stuff was all about!






BINGO! We have a winner!






We write: 


Slope
= 
Lim 
F(x +
Dx)
 F(x) 

DX
®
0 
Dx 






This limit is
the slope of the curvy line function at the point X. 








OK, Let's try
one for real, eh? 





Find the
derivative of: 


F(X) = X^{2} 





Break out the
new and improved slope formula and put X ^{2} 


(our F(X)
function value) into it: 





Slope
= 
Lim 
F(x +
Dx)
 F(x) 

DX
®
0 
Dx 






Slope
= 
Lim 
(x +
Dx)^{2}
 (x)^{2} 

DX
®
0 
Dx 






Stop a second
and make sure you see what just happened there. 





If the function
is F(X) = X ^{2} 


Then F(stuff) = (stuff) ^{2}



AND, AND,
AND, wait for it ... 





F( X + DX)
= ( X + DX)
^{2} 





Now, just
multiply this stuff out. 


Remember that X
and DX
are two completely different things. 


You CAN'T
combine them. 





Slope
= 
Lim 
x^{2} +
2xDx
+ Dx^{2}
 x^{2} 

DX
®
0 
Dx 
 




Combine terms. The X ^{2} and
X ^{2}
will cancel each other out.






Slope
= 
Lim 
2xDx
+ Dx^{2}


DX
®
0 
Dx 





And now for the sneaky part...



Factor a DX
out of the terms in the numerator.






Slope
= 
Lim 
(2x
+ Dx)Dx 

DX
®
0 
Dx 





Cancel the DX's 





Slope
= 
Lim 
(2x
+ Dx)Dx
^{1} 

DX
®
0 
Dx
^{1} 





Slope
= 
Lim 
2x
+ Dx 
DX
®
0 






Evaluate the
limit. 


Remember, X is
not the same thing as DX.
The limit only cares about DX.






Slope
= 2X + 
Lim 
Dx 
DX
®
0 






Slope
= 2X + 0 





Slope
= 2X 





Did you see how
the Limit just totally ignored the 2X? 





In algebra, you
might have used "m" to stand for the slope. 


Math people
seem to get bored real easy. 


They never
leave anything alone for long. 


So they came up
with lots of new names for the slope. 





They call it: 


Y' or F'(X) or 
dY 
or
The Derivative 

dX 



So from the
last example, we could say: 


Y' =
2X or F'(x) = 2X
or 
dY 
= 2X 

dX 



or 


The
Derivative is 2X 





Which of
these you use is usually pretty optional at this point. 


Some teachers
like one way or another and demand you use it... 


Like whatever
dude... 


Just do it
their way. 





If a line
curves, the derivative will be an equation. 


Since the
slope is different at different X values, 


we couldn't just have a
single number as the slope. 





To find the
slope of the original equation (F(X)
= X ^{2}) at any value
of X, 


just substitute
the X value you want the slope for into the derivative
equation. 


The answer you
get is the slope of F(X) = X ^{2} at that X value. 





Let's do
another one: 


Find the
equation of the slope (aka the Derivative) of: 





F(X) = X^{3}






Lim 
F(x +
Dx)
 F(x)

= 
Lim 
(x +
Dx)^{3}
 (x)^{3} 


DX
®
0 
Dx 
DX
®
0 
Dx 

Lim 
F(x +
Dx)
 F(x)

= 
Lim 
x^{3}
+ 3x^{2}
Dx +
3x Dx^{2}
+ Dx^{3}
 x^{3} 


DX
®
0 
Dx 
DX
®
0 
Dx 

Lim 
F(x +
Dx)
 F(x)

= 
Lim 
3x^{2}
Dx +
3x Dx^{2} +
Dx^{3} 


DX
®
0 
Dx 
DX
®
0 
Dx 

Lim 
F(x +
Dx)
 F(x)

= 
Lim 
(3x^{2}
+ 3x Dx +
Dx^{2})
Dx 




DX
®
0 
Dx 
DX
®
0 
Dx 


Lim 
F(x +
Dx)
 F(x)

= 
Lim 
3x^{2} +
3x Dx +
Dx^{2} 

DX
®
0 
Dx 
DX
®
0 

Lim 
F(x +
Dx)
 F(x)

= 
3x^{2} +
3x 
Lim 
Dx
+ 
Lim 
Dx^{2} 

DX
®
0 
Dx 
DX
®
0 
DX
®
0 

Lim 
F(x +
Dx)
 F(x)

= 
3x^{2} +
3x ´
0 + 0 

DX
®
0 
Dx 

Lim 
F(x +
Dx)
 F(x)

= 
3x^{2} 

DX
®
0 
Dx 




Did you
see? 


Any term
that had DX
in it went away when we did the
DX
®
0 stuff. 





In more
formal math type language: 


A term
that included DX
became 0 when we took the limit of the term 


when
DX
approached zero. 





We also
had more of that stuff x 0 = 0 shenaninans. 





Remember,
those terms just go away. 


When you
find them, beat the Christmas rush and start ignoring them early. 





copyright 2005 Bruce
Kirkpatrick 
