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### What is dx?

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Date: 08/25/2001 at 13:31:38
From: Michael Morse
Subject: dx

What does dx mean and where does it come from?

Thanks.
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Date: 08/26/2001 at 06:10:56
From: Doctor Jeremiah
Subject: Re: dx

Hi Michael,

I assume by "dx" you mean the calculus version of dx.

Calculus is all about how to measure the slope of any arbitrary line,
especially curved ones.

Consider y = 2x^2  (the x^2 means "x squared").

If you used the "normal" method to get the slope you would pick two
points (lets pick (1,2) and (3,18) for this example) and then you
would make a ratio of the "rise" (the difference in the y values) and
the "run" (the difference in the x values)

If you did this you would have a slope of:

m = (y2-y1)/(x2-x1) = (18-2)/(3-1) = 16/2 = 8

The problem with this method is that it produces the wrong answer.
The only time it's right is for a straight line. For example, pick two
different points: (2,8) and (3,18)

Then you would have this slope:

m = (y2-y1)/(x2-x1) = (18-8)/(3-2) = 10/1 = 10

A curved line is not straight, so the slope will never be right except
for one thing: the closer the two points are to each other, the more
accurate the slope is. The curve gets flatter and flatter as the two
points get closer and closer. When they get infinitely close to each
other we get the most accurate answer because essentially the points
are so close to each other that there is no room for any curvy bits.

If we define "dx" to be the difference between two x-values that are
infinitely close to each other (an infinitely small difference in x
values), and we define "dy" to be the difference between two y-values
that are infinitely close to each other (an infinitely small
difference in y values), then we can pick two infinitely close points
and do this:

m = (y2-y1)/(x2-x1) = dy/dx

So dy/dx is the slope of a line. If we use the rules of calculus to
"differentiate" our equation (using the mythical d function):

y = 2x^2
d(y) = d(2x^2)
dy = 2 d(x^2)
dy = 2*2x*d(x)
dy = 2*2x*dx
dy = 4x*dx

We find that an infinitely small difference in y can be measured with
this equation: dy = 4x*dx. But if we rearrange it slightly:

dy = 4x*dx
dy/dx = 4x*dx/dx
dy/dx = 4x*1
dy/dx = 4x

We find that the slope of y = 2x^2 is 4x.  Notice that the slope is
not a number; it actually changes depending on where in the graph we
are; you can see that the slope changes by graphing y = 2x^2.

So the slope at any point on the graph can be found with this equation
because the two points that we use to calculate with are infinitely
close together (for all intents and purposes they are the same point)

And since we know the definitions of dx and dy, we could say that the
slope at any point equals an infinitely small difference in y (dy)
divided by an infinitely small difference in x (dx). This is
absolutely true. And for a straight line graph it is the same as
taking the difference of any two points.

- Doctor Jeremiah, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Calculus

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