I just stumbled upon a simple notation inconsistency in partial derivatives and I'm unsure on how to make it appear more correctly. Assume
$f(x,y) := (x+y)*x$
For no particular reason, let's define
$u(x,y) := x + y$
so that we can rewrite $f(x,y) := u*x$. Now in Leibniz notation we could express the partial derivative of $\frac{df}{dx}$ by
$$\frac{df}{dx} = \frac{df}{df}\frac{df}{dx} + \frac{df}{df}\frac{df}{du}\frac{du}{dx} = 1*u + 1*x*1 = 2x + y$$
What bothers me is that i have $\frac{df}{dx}$ on the left hand side and the right hand side of the equation, but clearly there is a different meaning to both terms.
You've used some slightly erroneous notation. Recall that a partial derivative is the same as a total derivative when the respective variable is the only changing parameter.
The partial derivative for $f$ as you've defined it, with respect to $x$, is:
$$\dfrac{\partial f}{\partial x} = \dfrac{\partial}{\partial x} ux$$
Employing the product rule, we obtain:
$$x\dfrac{\partial}{\partial x} u+u\dfrac{d}{dx} x=x(1)+u(1)=x+u=2x+y.$$
Note that $f=(x+y)x=x^2+xy$ and taking the partial with respect to $x$ from here will give the same answer. Does this help?