I cannot understand something regarding chain rule for partial differentiation.
Let $f(x,y)$. If I understand correctly, according to the chain rule we get:
$\frac{\partial}{\partial u}f(x,y)=\frac{\partial f(x,y)}{\partial x}\cdot\frac{\partial x}{\partial u}+\frac{\partial f(x,y)}{\partial y}\cdot\frac{\partial y}{\partial u}$
Now, for $u=x$, we end up with the term $\frac{\partial f\left(x,y\right)}{\partial x}$ in both sides of the equality; aren't we getting here some kind of loop?
$\frac{\partial f\left(x,y\right)}{\partial x} =\frac{\partial f\left(x,y\right)}{\partial x}\cdot\frac{\partial x}{\partial x}+\frac{\partial f\left(x,y\right)}{\partial y}\cdot\frac{\partial y}{\partial x}= \frac{\partial f\left(x,y\right)}{\partial x}+\frac{\partial f\left(x,y\right)}{\partial y}\cdot\frac{\partial y}{\partial x}$