Implicit Differentiation: single-variable vs. multi-variable

Metronome

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I noticed that the implicit differentiation procedure from single variable calculus, i.e.,

x^3 + sin(y) = y ---> (3x^2)dx + (cos(y))dy = (1)dy ---> (3x^2)dx = (1 - cos(y))dy ---> dy/dx = (3x^2)/(1 - cos(y))) (better explanation:
https://www.youtube.com/watch?v=qb40J4N1fa4)

doesn't appear to work for implicit multivariable functions. The formula used in the multivariable world is...

dy/dx = -(∂z/∂x)/(∂z/∂y)

This is said to be derived from the multivariable chain rule scenario where z is a function of x and y, and y is a function of x, in other words, z(x, y(x)). I want to know how robust this formula is. Must the equation being implicitly differentiated have the z(x, y(x)) dependency structure, or does it work on any equation with any dependency interrelations? Can it be generalized to d1/d2 = -(∂3/∂2)/(∂3/∂1), where 1, 2, and 3 can be assigned the variables x, y, and z in any order, in case it is desirable to implicitly differentiate some derivative other than dy/dx? Does this formula connect with the single variable implicit differentiation procedure mentioned above, or is multivariable implicit differentiation just a different animal?
 
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I noticed that the implicit differentiation procedure from single variable calculus, i.e.,

x^3 + sin(y) = y ---> (3x^2)dx + (cos(y))dy = (1)dy ---> (3x^2)dx = (1 - cos(y))dy ---> dy/dx = (3x^2)/(1 - cos(y))) (better explanation:
https://www.youtube.com/watch?v=qb40J4N1fa4)

doesn't appear to work for implicit multivariable functions. The formula used in the multivariable world is...

dy/dx = -(∂z/∂x)/(∂z/∂y)

This is said to be derived from the multivariable chain rule scenario where z is a function of x and y, and y is a function of x, in other words, z(x, y(x)). I want to know how robust this formula is. Must the equation being implicitly differentiated have the z(x, y(x)) dependency structure, or does it work on any equation with any dependency interrelations? Can it be generalized to d1/d2 = -(∂3/∂2)/(∂3/∂1), where 1, 2, and 3 can be assigned the variables x, y, and z in any order, in case it is desirable to implicitly differentiate some derivative other than dy/dx? Does this formula connect with the single variable implicit differentiation procedure mentioned above, or is multivariable implicit differentiation just a different animal?

Here is an explanation of what implicit differentiation means in this context, http://tutorial.math.lamar.edu/Classes/CalcIII/PartialDerivatives.aspx, and a subsequent explanation (linked at the end of the first) of the method you are asking about, http://tutorial.math.lamar.edu/Classes/CalcIII/ChainRule.aspx#ImplicitDiffSingle . You will note that it is about implicit partial differentiation, not literally about dy/dx as you are saying. If nothing else, this shows the derivation that you don't appear to have seen, but only heard of.
 
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