TITLE: chain rule using tree diagram, why does it work?
QUESTION [11 upvotes]: In multivariable calculus, I was taught to compute the chain rule by drawing a "tree diagram" (a directed acyclic graph) representing the dependence of one variable on the others. I now want to understand the theory behind it.
Examples:
Let $y$ and $x$ both be functions of $t$.
Let $z$ be a function of both $x$ and $y$.
The derivative of z with respect to t is:
$$\frac{dz}{dt} = \frac{\partial z}{\partial x} \frac{dx}{dt} + \frac{\partial z}{\partial y} \frac{dy}{dt}$$
To compute this derivative, I was taught to draw a graph with the following edges:
$x \to z$, $y \to z$, $t \to x$, and $t \to y$.
Source: http://www.math.hmc.edu/calculus/tutorials/multichainrule/

These tree diagrams can be constructed for arbitrarily complex functions with many variables. 
In general, to find a derivative of a dependent variable with respect to an independent variable, you need to take the sum of all of the different paths to reach the dependent variable from the independent variable. Traveling down a path, you multiply the functions (e.g. $\frac{\partial z}{\partial x} \cdot \frac{dx}{dt}$).
Why does this work?

REPLY [2 votes]: This video will certainly clarify things: http://www.youtube.com/watch?v=2bF6H_xu0ao.
Although it may take a bit longer, I personally find that computing the total differential is substantially easier and more intuitive than a tree diagram.