Chain rule for multi-variable functions So I have been studying the multi-variable chain rule.  Most importantly, and this is what I must have overlooked, is it's not always clear to me how to see which variables are functions of other variables, so that you know when to use the chain rule.  For example, if you have:
$$ x^2+y^2-z^2+2xy=1 $$
$$ x^3+y^3-5y=8 $$
In general, say we want to find $\frac{dz}{dt}$ but $z$ is a function of $x$, then we get:
$$ \frac{dz}{dt} = \frac{dz}{dx} \frac{dx}{dt} .$$  
And if $z$ is a function of both $y$ and $t$, we get:
$$ \frac{dz}{dt} = \frac{dz}{dx} \frac{dx}{dt} + \frac{dz}{dy} \frac{dy}{dt}$$  
In this case, we have two equations.  One involving all three variables $x,y,z$ and one involving just $x,y$.  Say we want to find $\frac{dz}{dx}$. What does this mean for this case?  How should we interpret this rule in general? 

What you want to look up is the chain rule for multivalued functions.  What follows deals with the second example, where $z$ is a function of $x$ and $y$.
Suppose $z:\mathbb{R}^n\rightarrow \mathbb{R}$ and $\boldsymbol{T} : \mathbb{R} \rightarrow \mathbb{R}^n$ are two differentiable functions.  Then $z\circ \boldsymbol{T} : \mathbb{R}\rightarrow \mathbb{R}$ and is differentiable, so we can ask what is $\frac {d(z(\boldsymbol{T}(t)))}{dt}$?  It turns out that at $t=t_0$ we have $$\frac {d(z(\boldsymbol{T}(t)))}{dt} = \nabla z(\boldsymbol{T}(t_0))\cdot \boldsymbol{T}^'(t_0) $$  Where $\nabla$ is the gradient and the $\cdot$ refers to the dot product between vectors.  
Hope that helps,
