Notation Abuse for dependent variable differentiation Let $z(x,y,t) = x^2+y+t^2$ where $x(t)=t$.
$\left(\dfrac{\partial z}{\partial x}\right)_{y,t} = 2x$
If you substitute in x for a t, you get the following.
$z(x,y,t) = x^2+y+xt$
$\left(\dfrac{\partial z}{\partial x}\right)_{y,t} = 2x+t$
This type of issue only occurs when you take partial derivatives in respect to dependent variables.  I'm assuming a new function should be defined for all variable substitutions, where this sloppy notation tends to work out nice for differentiation in respect to independent variables?  Variable substitution within functions occurs a lot in engineering which leads to any possible recommendations to avoid this?
 A: The notation $\left(\dfrac{\partial z}{\partial x}\right)_{y,t}$
describes what happens to $z$ when you vary $x$ while holding $y$ and $t$ fixed.
Here's the catch: you cannot (1) vary $x$ while (2) holding $t$ fixed if 
(3) the value of $x$ is a function of $t.$
Those three things are mutually exclusive.
In order to evaluate $\left(\dfrac{\partial z}{\partial x}\right)_{y,t},$
you have to suspend the premise that $x$ is a function of $t.$
In fact, you must treat $x$ and $t$ as independent variables, at least within
a neighborhood of the point where you are evaluating the partial derivative.
In practice, you can run into a situation like yours when there is a physical formula
that relates $t,$ $x,$ $y,$ and $z$ in a general set of circumstances,
but your particular set of circumstances impose a constraint on $x$ and $t$
that is not present in the general case.
If the constraint were always present, it would have been incorporated into
the general formula, which would have been a function of fewer variables, 
such as $z(y,t).$
I think you have two choices in such a case. One is to write a new formula
(with fewer independent variables) that applies to your specific set of
circumstances but not to the general case, and use that formula exclusively.
The other is to compute all your partial derivatives under general-case assumptions,
where $x$ is not a function of $t.$  When you come to compute total derivatives
(possibly with the help of the partial derivatives you found), 
you can apply the constraint.
