Derivation of Flory's Model for Linear Condensation of Polymers

Question

I'm having some trouble understanding a part of Flory's model for linear polycondensation reactions. In his 1936 paper, he derives an expression for the probability of the existence of a particular $x$-mer as

$$\Pi_x = xp^{x-1}(1 - p)^2\tag{2}$$

Later, he states the following:

In order to locate the $x$ value which gives a maximum in $\Pi_x,$ let

$$\partial\Pi_x/\partial x = (1 - p)^2(p^{x - 1} + xp^{x - 1}\ln p) = 0$$

Not being the most mathematically savvy person, can someone explain to me how he got to this expression?

Reference

1. Flory, P. J. Molecular Size Distribution in Linear Condensation Polymers. J. Am. Chem. Soc. 1936, 58 (10), 1877–1885. DOI: 10.1021/ja01301a016.

Answer 1

Given $\Pi_{x}$= $𝑥*𝑝^{𝑥−1}$ *$(1−𝑝)^2$

$\partial$ refers to partial differentiation.

$\frac{\partial \Pi_{x}}{\partial x}$ means differentiating $\Pi_{x}$ with respect to x.

Therefore $(1−𝑝)^2$ is a constant when differentiating with respect to x. So during differentiation the constant stays as a constant multiplier.

Now see that there are two terms involving x, therefore we need to use the product rule of differentiation.

$\left(\frac{\partial \Pi_{x}}{\partial x}\right)$ = $(1−𝑝)^2$ * ($𝑝^{𝑥−1}* \frac{\partial x}{\partial x} + x*\frac{\partial p^{x-1}}{\partial x}$)

$\frac{\partial x}{\partial x}$ = 1 and for calculating $\frac{\partial p^{x-1}}{\partial x}$ look at exponentiation rule of differentiation as suggested by @Ian Bush in the comments.

Therefore we finally have :

$\left(\frac{\partial \Pi_{x}}{\partial x}\right)$ = $(1−𝑝)^2$ * ($𝑝^{𝑥−1}* 1 + x*ln(p)*p^{x-1}$)

And since he is trying to locate the 𝑥 value which gives a maximum in $\Pi_{x}$, he equates the equation to 0.

$\left(\frac{\partial \Pi_{x}}{\partial x}\right)$ = $(1−𝑝)^2$ * ($𝑝^{𝑥−1}* 1 + x*ln(p)*p^{x-1}$) = 0