Let $G(n, p)$ be the Erdős–Rényi random graph model with $n$ vertices and $p\in[0,1]$. Furthermore let $\mathbb{E}_{n,p}[k]$ be the average number of clusters (counting isolated vertices as $1$). I was wondering if it is possible to say something about the asymptotics of
$$ p \frac{1}{n}\frac{d}{dp} \mathbb{E}_{n,p}[k]$$
Especially evaluated at (close to) $pn=1$ I wonder about
$$\lim_{n\rightarrow \infty} \frac{1}{n^2} \frac{d}{dp} \mathbb{E}_{n,p}[k]$$
Will it tend to zero?
Given $n$ vertices, there are $f_{n,k}$ groups of $k$ vertices, where $f_{n,k} = \binom{n}{k}$. A particular group of $k$ vertices is a $k$-cluster (i.e. every couple of vertices in the group is connected) with probability $g_{k,p} = p^k$. The probability that there are exactly $x$ $k$-clusters is:
$$ h_{n,k,p}(x) = \binom{n}{x}g_{k,p}^x(1-g_{k,p})^{n-x} $$
The expected value of $x$ is $\mathbb{E}_{n,p,k}[x]$. Since $x$ is a binomial random variable, then
$$\mathbb{E}_{n,p,k}[x] = ng_{k,p} = np^k$$
The expected value of the total number of clusters is:
$$\mathbb{E}_{n,p}[k] = \sum_{x=1}^n x\mathbb{E}_{n,p,k}[x] = \sum_{x=1}^n nxp^k = np^k \sum_{x=1}^n x = np^k \frac{n(n+1)}{2} = p^k \frac{n^2(n+1)}{2}$$
Moreover:
$$\frac{\text{d}\mathbb{E}_{n,p}[k]}{\text{d}p} = kp^{k-1}\frac{n^2(n+1)}{2}$$
You can go on now.
Note: some edits added