I am wondering if there is symmetry among the single partial derivative of a multivariate diffeomorphic function.
Let $f: \mathcal{X} \rightarrow \mathcal{X}$ be a diffeomorphism defined over $\mathcal{X} \subset \mathbb{R}^d$ (it is a d-dimensional multivariate function). Let $\mathbf{V} \in \mathcal{X}$ be some d-dimensional set of variables.
Say we have the following constraint on one of the single partial derivatives:
$\frac{\partial V_i}{\partial f_{i}(\mathbf{V})} = 0$, does this imply that $\frac{\partial f_i(\mathbf{V})}{\partial V_i} = 0$?
If so, is there a name for such a result? If this does not hold for a diffeomorphism, then are there any general conditions that would make this property hold?
The closest thing I could find was something about the symmetry of mixed partial-derivatives (i.e. taking two partial derivatives) Symmetry of mixed partial derivatives..