In a problem I was working on, I found it convenient to use the notation $ dX/dA_{i \rightarrow j} $ to represent the marginal change in $X$ from redistributing a marginal amount of $A_i$ to $A_j$. Is there a name for this, and can it even be called a "derivative"? Is there a better or more conventional way to write this?
Context: $\{ A_i \}$ is a finite sequence of loan payments and $X$ could be something like the associated internal rate of return. There are many valid payment sequences that satisfy a set of constraints, each having a different $X$.
If I understand the question correctly, I believe that this falls under the concept of directional derivative.
In particular, if we treat $X$ as a function of the $A_i$, e.g.: $X(A_1, A_2, \ldots, A_n)$, then the quantity described as $dX/dA_{i \rightarrow j}$ could be described equivalently as $\nabla_v X$ where $v$ is some vector that is positive in the $A_j$ direction, negative in the $A_i$ direction, and zero elsewhere.
EDIT:
To make it even more explicit in this case, a redistribution represented by $A_{i \rightarrow j}$ corresponds to a situation where a one unit decrease in $A_i$ is matched by a $(1+r)^{j−i}$ increase in $A_j$. Then, we can define $u$ to be a vector with $-1$ in the $i$th place, $(1+r)^{j−i}$ in the $j$th place, and zero elsewhere. Define $v = u/||u||$ (i.e.: $v$ is $u$ normalized to unit length). Then, $dX / dA_{i \rightarrow j} = \nabla_v X$.