Hi! I'm logged in now. Take a look at this page (http://www.math.ucdavis.edu/~kouba/CalcTwoDIRECTORY/usubdirectory/) and start reading the section from the words "Now the method of u-substitution will be illustrated on this same example. Begin with"

The sentence "Now "pretend" that the differentiation notation" (the 4th sentence down from "Now the method of u-substitution...") is the part you're getting stuck at? Hm. Maybe if I wrote your example this way it'll help you understand.

Let Y = 1+X
dY/dx = 1
(dY/dx)*(dx) = (1)*(dx)
(dY)*(dx/dx) = (1)*(dx)
dY = 1 dx (because dx/dx = 1 )

Do you see how I got from the 2nd equation (dY/dx = 1) to the last equation (dY = 1 dx)?

Now instead of Y, we say U

Let U = 1+X
dU/dx = 1
(dU/dx)*(dx) = (1)*(dx)
(dU)*(dx/dx) = (1)*(dx)
dU = 1 dx

Are you okay with this so far?