How does $F(X_1,X_2)$ change with $\epsilon$ change in $X_1$ when $X_1$ and $X_2$ are entangled? I have the following function $F$
$$F(X_1,X_2)=\frac{X_1}{X_1+X_2}$$
Where $X_1$ and $X_2$ are functions of a variable $k$. I'm trying to investigate how $F$ changes with a $\epsilon$ change in $X_1$. However, I cannot simply do
$$F(X_1+\epsilon,X_2)=\frac{X_1+\epsilon}{X_1+\epsilon+X_2}$$
Since if $X_1$ changes (due to a change in $k$) $X_2$ will also change.   
Hence, it makes sense (to me) to do the following. Define, $A:=\frac{dX_1}{dk}$ and $B:=\frac{dX_2}{dk}$. Then we have 
$$F(X_1+\epsilon,X_2)==\frac{X_1+\epsilon}{X_1+\epsilon+X_2+\epsilon(B/A)}$$
For instance if $B=2A$, then a change of $\epsilon$ in $X_1$ will come with a $2\epsilon$ change in $X_2$. Does this make sense?
 A: You might really just want to compute $h’(k)$ for $h$ defined: 
$$ h(k) = F(X_1(k), X_2(k))$$ 
where $X_1$ and $X_2$ are given functions of $k$. 
Or, if you have some (perhaps local) inverse function $X_1^{-1}(x_1)$ 
that is differentiable then you can find $g’(x_1)$ for $g$ defined: 
$$ g(x_1) = F\left(x_1, X_2(X_1^{-1}(x_1))\right)$$
This is related to other approaches that may seek to define  "$\frac{\partial X_2}{\partial X_1} = \frac{X_2’(k)}{X_1’(k)}$" and use it in a formula such as
$$ "\frac{\partial h}{\partial X_1} = \frac{\partial F}{\partial X_1} + \frac{\partial F}{\partial X_2}\frac{\partial X_2}{\partial X_1}" $$
A: Use the differential. Given $F: \mathbb{R}^2 \to \mathbb{R}$ which is at least $\mathcal{C}^1$ we have $dF(p) = \frac{\partial F}{\partial x}(p) \ dx + \frac{\partial F}{\partial y}(p)\ dy$ where $|\Delta F(p) - dF(p)|< \epsilon$ and $p$ is sufficiently close to $\textbf{x}$. Therefore to estimate an $\epsilon$ change in $X_1$ we have $dx =( X_1 + \epsilon) - X_1 = \epsilon, dy = 0$ and so,
$$ dF(p) = F_x(X_1,X_2)  \cdot \epsilon + Fy(X_1,X_2) \cdot 0$$
Hence by the above remark $F(X_1 + \epsilon, X_2) \approx F(X_1,X_2) + F_x(X_1,X_2) \cdot  \epsilon$. Here to not abuse notation I am writing $F = F(x,y)$ and so when we restrict to the set on which $F$ in your problem is defined (say $E$), we use the convention $(x,y)|_E = (X_1,X_2)$.
