Possible Lack of Rigor in Common Multivariable Chain Rule Expression? I'm trying to get my head around an issue with the multivariable chain rule as it's commonly written. By way of illustration, let's use this simple example, taken from here (about halfway down the page). $z$ is a bivariate function of $x,y$, ie $z=f(x,y)$. $y$ is also a function of $x$, ie $y=g(x)$. 
Now, we have this expression for the total derivative, or chain rule: $$\frac{dz}{dx}= \frac{\partial z}{\partial x} + \frac{\partial z}{\partial y} \frac{dy}{dx}      (1)$$
The meaning of $\frac{\partial z}{\partial y}$ is the rate of change of $z$ when $y$ is varied but $x$ is held constant. But since $y=g(x)$ holds at the same time, this is impossible to do, because if $x$ is held constant, so must be $y$ - it's impossible to vary $y$ and for $y=g(x)$ to hold at the same time. 
Of course, we know how to evaluate $\frac{\partial z}{\partial y}$: my way is to "pretend" that we don't know there is a dependence $y=g(x)$, and compute $\frac{\partial z}{\partial y}$ as if $x$ and $y$ are independent.
So:
Is Equation $(1)$, in fact, not rigorous? Is $y$ and $x$ doing "double duty"? If so, I'm trying to figure out how to make it rigorous - can "dummy variables" be introduced to do this?
 A: It is of course not rigorous. There are two functions mixed up here. One is 
$$ f \, : \,\,(x_1,x_2) \mapsto f(x_1,x_2),$$
the other is 
$$ z \, : \,\,x \mapsto f(x,g(x)).$$
So the statement $(1)$ should be like this:
$$\left.\frac{dz}{dx}\right\rvert_{x_0} = \left.\frac{\partial f}{\partial x_1}\right\rvert_{(x_0,g(x_0))} + \left.\frac{\partial f}{\partial x_2}\right\rvert_{(x_0,g(x_0))} \cdot \left.\frac{dg}{dx}\right\rvert_{x_0}  \qquad \cdots \quad    (1)$$
or, if you don't like the $x$ appearing all the time, like this :
$$\left.\frac{dz}{dx}\right\rvert_{x_0} = \left.\partial_1 f\right\rvert_{(x_0,g(x_0))} + \left.\partial_2 f\right\rvert_{(x_0,g(x_0))} \cdot \left.\frac{dg}{dx}\right\rvert_{x_0}  \qquad \cdots \quad    (1)$$
The notations $\partial_1 f$ and $\partial_2 f$ mean the partial derivatives of $f$ on the first and second variable.
A: We all have
$$\frac{dz}{dt}=
\frac{\partial z}{\partial x}\frac{dx}{dt} 
+\frac{\partial z}{\partial y}\frac{dy}{dt},$$
for any election of variables and dependences.
Now if we set $x=t$ we naturally get
$$\frac{dz}{dx}=
\frac{\partial z}{\partial x}\frac{dx}{dx} 
+\frac{\partial z}{\partial y}\frac{dy}{dx},$$ 
which upon simplification gives
$$\frac{dz}{dx}=
\frac{\partial z}{\partial x}
+\frac{\partial z}{\partial y}\frac{dy}{dx},$$
which is perfectly valid.
