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

(A) implicit differentiation: given f(x, y(x)) = f(x,y) =(x^3)(y^2) = c for a constant c, then dy/dx = -3y/2x
(B) partial differentiation: given f(x,y) =z=(x^3)(y^2) , then δf/δx = 3(x^2)(y^2) & δf/δy =2(x^3)y so doing something that is not allowed, (δf/δx)/(δf/δy) = δy/δx = 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? [My own impression is that it shouldn't be, but perhaps there is some better link between implicit and partial differentiations that someone can point out.]

 

[jedishrfu]

For partial differentiation the x and y are independent variables, but for the example you've shown y is dependent on x.

Why not try another function like f(x,y)=sin(x^2 + y^2) and see if it holds up?

 

[nomadreid]

right you are, jedishrfu. Thanks.