are implicit and partial differentiations related?

[nomadreid]

I was working two different but superficially related problems, and noticed that if I did something that is generally not allowed, the results were connected by a negative sign. My questions are whether this will always turn out this way, and if so, why.
The two problems were

1. implicit differentiation: given f(x, y(x)) = f(x,y) =(x^3)(y^2) = c for a constant c, then dy/dx = -3y/2x
2. partial differentiation: given f(x,y) =z=(x^3)(y^2) , then \(\displaystyle \delta\)f/\(\displaystyle \delta\)x = 3(x^2)(y^2) & \(\displaystyle \delta\)f/\(\displaystyle \delta\)y =2(x^3)y so doing something that is not allowed, (\(\displaystyle \delta\)f/\(\displaystyle \delta\)x)/(\(\displaystyle \delta\)f/\(\displaystyle \delta\)y) = \(\displaystyle \delta\)y/\(\displaystyle \delta\)x = 3y/2x.

Coincidence, or can this be generalized (that dy/dx = the negative of working with the partial derivatives in this way), and the sloppiness justified?

 

[tkhunny]

No, no, no, and no. Partial differentiationi requires two independent variables. Implicit differentiation assumes one varable is dependent. There is no other relationship assumed. Any odd relationship you may find must be purely coincidental or probably of very limited use.

 

[nomadreid]

thanks

Thanks, tkhunny, you're right, of course; mea culpa, I should have done more examples. (Coincidences are nice but usually meaningless, of course, as any astrologer will tell you.)