If we have function $y=L(x\_1,x\_2,x\_3,...,x\_n)$, and function $z=R(x\_1,x\_2,x\_3,...,x\_n)$. How to compute the derivative $\frac{dy}{dz}$?


Shall I do $\frac{dy}{dz} = \sup\_{g\in \Re^n}\frac{\bigtriangledown\_x L \cdot g}{\bigtriangledown\_x R \cdot g}$?


Is there any mathematical term associated with this kind of derivatives?



Implicit derivative?
There is no reason to expect that such a thing as $dy/dz$ exists. If you rephrase everything in terms of differentials, you have $$dy=\sum\_{i=1}^n\frac{\partial L}{\partial x\_i}dx\_i,\quad dz=\sum\_{i=1}^n\frac{\partial R}{\partial x\_i}dx\_i$$ where $dx\_1,\ldots,dx\_n$ are linearly independent, and so you find $dy=a\, dz$ if and only if $\nabla L=a\nabla R$. But normally, the two gradient won't be parallel, so you cannot define $dy/dz$. You *can* do so in any given direction, though: Your fraction $(\nabla L\cdot g)/(\nabla R\cdot g)$ is a perfectly good expression for $dy/dz$ *as measured in the direction given by $g$*.


