Let $f_q(x)$ be the generating function of the sequence $(q;q)_n$: $$f_q(x):=\sum_{n=0}^{\infty} (q;q)_n x^n,$$ where $(q;q)_n: = (1-q)(1-q^2) \cdots (1-q^n)$ with convention $(q;q)_0$:=1.

Let $g_q(x):=1/f_q(x)$.

Question: Are there closed form expressions for $$\lim_{x \rightarrow 1} g'_q(x) \quad \mbox{and} \quad \lim_{x \rightarrow 1} g''_q(x) \quad ?$$ By a rather tricky argument, I get $\lim_{x \rightarrow 1} g'_q(x) = -(q;q)_{\infty}$. But I don't get anything simple for $\lim_{x \rightarrow 1} g''_q(x)$.