Proving the Monotonicity of a function? Given a function of numerous variables, say $f(x,y,z)$, what are the usual approaches one can take to prove that $f(x,y,z)$ is monotonically increasing, or decreasing in $x$?
I am aware that one can calculate the functions derivative and attempt to prove that it is positive or negative for any $y$ and any $z$. However sometimes the derivatives are quite complex, and challenging to analyze.
Is there any other way to prove the monotonicity of a function?

It depends on how your function is defined. Many times proving directly is the easiest. 
i.e. Arbitrarily fix $x_1 < x_2$ and $y_{*}, z_{*}$ in your spaces $X$, $Y$, $Z$ that compose the domain of your function $X\times Y\times Z$. Show that $f(x_{1},y_{*},z_{*})<f(x_{2},y_{*}, z_{*})$. (same as what Jika said)
However, sometimes functions are defined in weird ways or they have many terms and it is easier to do something like this:
"Show that for any $x_{o}\in\mathbb{R}$, the function $$g_{x_{o}}(x)=f(x,y_{*},z_{*})-f(x_{o},y^{*},z^{*})$$ is only positive on $(x_{o}, \infty)$."
This method is particularly nice if it turns out that when you subtract the functions, you get something readily recognizable, like the power series expansion for a trig function—in which case, you might know something about whether $g_{x_o}$ is positive on $(x_{o}, \infty)$.
