Book example (and everywhere else online):
Differentiate the folium of descartes:
x^3 + y^3 = 6xy
Left side: 3x^2 + 3y^2 y'
Right side using product rule: 6xy' + 6y
Where did the x' go? (Ignoring the 6 for now):
The product rule is (fg)' = fg' + gf'
(xy)' = xy' + yx'
"xy' + yx'" is not "xy' + y" -- Where the heck does the x' go?
From the book example, it would have to be 1
Also, treating d/dx (x) as a fraction would be dx/dx = 1
I am not all that clear on why this is though. It seems like it should be terribly obvious to me, but I am unclear on just what "the derivative of x with respect to x" really means, particularly the "with respect to" part.
Is there a way to say "d(x) with respect to q" in Newton's prime notation?
Thankyou for your time, even if you don't make another reply!