Calculus Limit Method of Differentiation
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The Long Way
Limit Method of Differentiation

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 = Y2 - Y1 = 0 - (-2) = 2 = 2



X2 - X1 1 - 0 1
 
You can chose either point as point 1 (X1,Y1) 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(x2) - F(x1)

X2 - X1
 
 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 Y2 in disguise!"
 

Take the old "using two points to find the slope" equation:
 
Slope =  Y2 - Y1

X2 - X1
  
 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) = X2

 
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 x2 + 2xDx + Dx2 - x2

DX 0 Dx
 
Combine terms. The X 2 and -X 2 will cancel each other out.
Slope =   Lim 2xDx + Dx2

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) = X3

  
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 x3 + 3x2 Dx + 3x Dx2 + Dx3 - x3


DX 0 Dx DX 0 Dx
 
Lim F(x + Dx) - F(x) = Lim 3x2 Dx + 3x Dx2 + Dx3


DX 0 Dx DX 0 Dx
 
Lim F(x + Dx) - F(x) = Lim (3x2 + 3x Dx + Dx2) Dx


DX 0 Dx DX 0 Dx
 
Lim F(x + Dx) - F(x) = Lim

  3x2 + 3x Dx + Dx2


DX 0 Dx DX 0
 
Lim F(x + Dx) - F(x) = 3x2 + 3x  Lim Dx +  Lim

Dx2


DX 0 Dx DX 0 DX 0
 
Lim F(x + Dx) - F(x) =

3x2 + 3x 0 + 0


DX 0 Dx
 
Lim F(x + Dx) - F(x) =

 3x2


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

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