I think this is a common question in applied math but I find no occurrence of it in MSE.
$u=u(x), v=v(x)$
$f=f(u(x),v(x))$
1) $u$ and $v$ are both functions of the variable $x$. If $u$ varies, then that must be because of some variation in $x$, which in turn means that $v$ must also have varied. Is my logic correct thus far?
2) If (1) is correct, then there can be no variation in $u$ without a variation in $v$. A partial derivative of $f$, say $\frac{\partial f}{\partial u}$ would imply that $v$ is constant while $u$ varies. Isn't that a mathematical contradiction?
Thanks