How to derive the term acting like a first derivative with respect to A that I found by accident?

Dear community,

I would like to analyze how the following function changes, when variables A,B, or C are not zero:

$f(A,B,C) = \frac{Q(1+X)(PA-VB)-FC}{GX}-\frac{A(1+X)(QP(X+A+AX)-QV(X+B+BX)-FC)}{GX(X+A+AX)}$

with

$A,B,C,Q,P,V,F,G∈R$

All other variables besides $A,B,C$ are relevant for the calculation but simply not of interest for my analysis.

Note that $A,B,C$ being zero in the initial state actually leads to $f(A=0,B=0,C=0)=0$, so it holds that

$∆f(A,B,C) =f(A,B,C)-f(A=0,B=0,C=0)= f(A,B,C)-0 = f(A,B,C)$

and the change in $f(A,B,C)$ is generally given by $f(A,B,C)$ itself.

 
My problem:

Now I noticed by accident in Excel that I can also calculate the value of $∆f(A,B,C)$ equaling $f(A,B,C)$ for any parameter set by

$∆f(A,B,C)=f(A,B,C)=\frac{QV(1+X)}{G(X+A+XA)}*A-\frac{QV(1+X)}{G(X+A+XA)}*B -\frac{F}{G(X+A+XA)}*C$

You can try it yourself by putting in any numbers, the initial equation and this one always have the same result.

Note that this structure is very helpful, because the second and third term are the constant first derivatives with respect to $B$ and $C$, so I can also derive them separately and discuss their effects in my analysis:

$∆f(A,B,C)=f(A,B,C)=\frac{QV(1+X)}{G(X+A+XA)}*A+\frac{df}{dB}*B +\frac{df}{dC}*C$

Questions:

1) The first term in the last function is not $\frac{df}{dA}*A$, because $\frac{df}{dA} = \frac{(1+X)(QV(X+B+XB)+FC)}{G(X+A+XA)^2}$ acc. to an online calculator and is non-constant. Why is this term effectively still working as a "sort of derivative with respect to A", while not actually being the first derivative? Is this a special form of derivative, maybe considering that A is initially zero, that I can derive somehow? How?

2) If deriving the first term as a sort of derivative is not possible: How can I at least transform the initial function step-by-step to reach the shown last form?

Appendix:
Formula in linear form: (Q*(1+X)*(P*A-V*B)-F*C)/(G*X)-(A*(1+X)*(Q*P*(X+A+A*X)-Q*V*(X+B+B*X)-F*C))/ (G*X*(X+A+A*X))