What is dx?

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

What does dx mean and where does it come from?

Thanks.

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/