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Let w=f(x,y,z)=2x+3y+4z, which is defined for all (x, y, z) in R3. Suppose that we are interested in the partial derivative wx on a subset of R3, such as the pane P given by z=4x-2y. The point to be made is that the result is not unique unless we specify which variables are considered independent.
Repeat the arguments of parts (a) and (b) to find (∂y∂w), (∂y∂w)z,(∂z∂w)x], and (∂z∂w)y.