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?
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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