Suppose I have two variables, $a$ and $b$, that are related by $a + b = 1$. This establishes that the only acceptable values of $a$ and $b$ are on a line in the $(a,b)$ plane from $(0,1)$ to $(1,0)$.
Consider a function $f(a,b)$. Can we compute the partial derivative, $\partial f / \partial a|_b$, even though making a differential change of $a$ while $b$ is fixed will fail to satisfy $a + b = 1$?
I think we can't compute the derivative, because fixing $b$ fixes $a$. In my field it appears common to simply ignore algebraic constraints between variables - that is, computing $\partial f / \partial a$ and $\partial f / \partial b$ as if there is no dependency between $a$ and $b$. This massively simplifies the calculation, but I think it's wrong.
Context: computing sensitivities of a differential-algebraic system governing the evolution of chemical reactors. The constrained variables are chemical mass fractions (must sum to 1), and the functions of interest are reaction rates. The above is an attempt at a minimum working example.