Problem understanding an implicit differentiation

Question

Here is a general budget constraint: $p{_1}x{_1}+p_{2}x_{2}=M\Leftrightarrow \frac{p_1}{p_2}x_1+x_2=\frac{M}{p_2}\Leftrightarrow {p_{1}}'x_1+x_2=M{}'$.

The main idea is that since prices are given, we can choose ${p_{2}}'=1$. However, I don't quite understand how they went on to the final line of the budget constraint from the second line.

I reckon they did implicit differentiation, but I am not understanding how they did it.

Edit: The $p's$, $x's$, and $M$ are price, quantity, and income, respectively.

Answer 1

$M=M(p_1,p_2,\overline U)$: The minimum income needed to buy the quantities of commodity 1 and commodity 2 that gives utility $\overline U$.

$M=p_1 \cdot x_1 + p_1 \cdot x_2 \Rightarrow p_1=\frac{M(p_1,p_2 , \overline U)-p_2 \cdot x_2}{x_1}$

$\frac{\partial p_1}{\partial p_2}=\frac{M'-x_2}{x_1}\Rightarrow p_1'\cdot x_1=M'-x_2$

Thus $p_1'\cdot x_1+x_2=(M'-x_2)+x_2=M'$

Answer 2

When a derivative is written, it represents an instantaneous rate of change of some dependent variable with respect to another (independent) variable. In your equation, it is not clear what the variable of differentiation might be; neither is it clear how the other quantities might depend on it.

I suppose such a lack of notational precision and mathematical rigor should not come as a surprise given that the context is economic theory.

My best guess about how they went from the second equation to the third (rightmost) equation is that they are using some kind of infinitesimal argument to write $p_1/p_2 \approx dp_1/dp_2$ and $M/p_2 \approx dM/dp_2$. I don't know what these quantities represent, so without further clarification I see no point in further speculation.