Suppose we have a series expansion for $f(x):=\sum_{k\geq 0}(-1)^ke^{-(2k+1)^2x}$ in the neighborhood of $0$, that is, $f(x)=\sum_{n\geq 0}a_n x^n$ for $|x|$ small. I want to know the value of $a_1=f'(0)$. Let $z=e^{-x}$, the concerned function is then $g(z):=\sum_{k\geq 0}(-1)^k z^{(2k+1)^2}$. And it suffices to calculate $g'(1)$.

Anybody knows how to find $f'(0)$ or $g'(1)$, or knows some identity for $f$ or $g$ to make the calculation easier?