Firstly, you can't just push the parts of a (partial) derivative $\partial u/\partial x$ around as you like: $\partial u/\partial x$ is one quantity, that happens to be composed of a number of symbols. This is clearer if you write it as $u_x$ or $u_{,x}$.
The next problem is that while for total derivatives or functions of one variable
$$ \frac{dy}{dx} \frac{dx}{dy} = 1, $$
the same is not the case for partial derivatives, as the following example will illustrate: let
$$ u = x+y, \qquad v=x-y. $$
Then inverting gives
$$ x = \frac{u+v}{2} \qquad y = \frac{u-v}{2}. $$
So,
$$ \frac{\partial u}{\partial x} = 1 \qquad \frac{\partial u}{\partial y} = 1 \\
\frac{\partial x}{\partial u} = \frac{1}{2} \qquad \frac{\partial y}{\partial u} = \frac{1}{2},
$$
and therefore
$$ \frac{\partial u}{\partial x} \frac{\partial x}{\partial u} = \frac{1}{2} \qquad \frac{\partial u}{\partial y} \frac{\partial y}{\partial u} = \frac{1}{2}, $$
so the sum is $1$, but the products are not $1$ individually.
A further instructive example is the even simpler-looking example
$$ u = x + ay, \qquad v = y. $$
Then
$$ x = u-av \qquad y = v. $$
So $ \frac{\partial v}{\partial x} = 0 $, but
$$ \frac{\partial x}{\partial v} = -a. $$
This tells us two interesting things: firstly, what we already know about the product not being $1$. And secondly, that although $y=v$, $\partial x/\partial y = 0 $ ($x$ and $y$ are meant to be independent to start with, after all!), so $ \frac{\partial x}{\partial v} \neq \frac{\partial x}{\partial y} $. What does this mean? It means that the partial derivative very much depends on what is being held constant, in addition to what is allowed to vary.
In the case of $\frac{\partial x}{\partial y}$, the coordinates other than $y$ (namely $x$) is being held constant, so of course this gives zero. But in $\frac{\partial x}{\partial v}$, $u$ is being held constant, and since this is not the same as $x$, the answer is different. This is one of the most difficult, and most important, things to understand about partial derivatives. There's a nice illustration of this on p.190 of Penrose's Road to Reality.
The correct approach is to apply the chain rule for partial derivatives, namely that if $f$ is a function of $x,y,z$, which in turn are functions of $u$ and other variables, then
$$ \frac{\partial f}{\partial u} = \frac{\partial x}{\partial u}\frac{\partial f}{\partial x} + \frac{\partial y}{\partial u}\frac{\partial f}{\partial y} + \frac{\partial z}{\partial u}\frac{\partial f}{\partial z}. $$
If then $f=u$, the result follows.
