derivative on both sides I am reading about feedback topologies and having some problems about math. I am not a math student so I need your help. Could you explain if the operation of taking derivative of both sides correct rigorously. 
I read that we can't treat dA as a number (standard analysis) and there is something about differential form that could make it correct.
Am I understanding it correctly?

 A: Unles you Really Need to Know More, it's probably best to view the calculation as follows:


*

*The equation
$$
A_{f} = F(A) = \frac{A}{1 + A\beta}
\tag{1}
$$
represents a functional dependence in the sense that knowledge of the value of $A$ uniquely determines the value of $A_{f}$.

*The derivative of $F$,
$$
F'(A) = \lim_{h \to 0} \frac{F(A + h) - F(A)}{h} = \frac{1}{(1 + A\beta)^{2}}
\tag{2}
$$
has a well-defined mathematical existence.

*The derivative of $F$ at $A$ is the linear coefficient of an approximation
$$
F(A + h) = F(A) + h\, F'(A) + E(h),\qquad \lim_{h \to 0} \frac{E(h)}{h} = 0.
\tag{3}
$$
(The second equation effectively says "the error in assuming $F(A + h) = F(A) + h\, F'(A)$ is negligible compared to $h$ when $h$ is small".)

*In terms of the variables in the functional dependence (1), equation (3) reads
$$
\underbrace{A_{f} + dA_{f}}_{F(A + h)} = \underbrace{A_{f}}_{F(A)} + h\, F'(A) + E(h).
\tag{4a}
$$
Viewing $h = dA$ as an increment in $A$ small enough that $E(dA)/dA$ is effectively $0$, equation (4a) may be written (after cancelling the $A_{f}$'s)
$$
dA_{f} = F'(A)\, dA,\quad\text{or}\quad
F'(A) = \frac{dA_{f}}{dA}.
\tag{4b}
$$
The latter is "Leibniz notation" for a derivative.

*By (4a), we may view (4b) as exact in the limit as $dA \to 0$, and (with negligible error, for some definition of "negligible") may view $dA_{f}$ and $dA$ as small numerical quantities, for which
$$
dA_{f} = \frac{dA_{f}}{dA}\, dA,\qquad
\frac{dA_{f}}{A_{f}} = \frac{1}{1 + A\beta}\, \frac{dA}{A},
$$
and so forth.
In summary, if two real variables $x$ and $y$ are related by a functional dependence $y = F(x)$, then $F'(x) = dy/dx$ may (for many purposes) be treated algebraically as a ratio of infinitesimal increments. Particularly, if $dx$ is "sufficiently small", then $dy = F'(x)\, dx$ up to a "negligible" error.
A: You take the derivative of $A_f$ w.r.t $A$.
$A_f=\frac{A}{1+A \beta }$
Taking derivative of $A_f$. Here you apply the quotient rule:
$\frac{d}{dx}\frac{f(x)}{g(x)} =\frac{f'g-g'f}{g^2}$
$\frac{dA_f}{dA}=\frac{1\cdot (1+A \beta)-\beta\cdot A}{\left(1+A \beta \right)^2}$
The numerator on the RHSright side becomes 1.
$\frac{dA_f}{dA}=\frac{1}{\left(1+A \beta \right)^2}$
$dA$ is the differential. The value of $dA$ tends to zero, but it is not zero. Therefore you can multiply the equation by $dA$. 
