Let $f(z)=\lambda(-\frac{1}{1+z})=\frac{1}{1-\lambda(z)}$, where $\lambda(z)$ is the modular lambda function, $\lambda(z)=16q-128q^2+704q^3+\ldots$, where $q=e^{\pi i z}$ (https://en.wikipedia.org/wiki/Modular_lambda_function).
What is the growth rate of the values of higher order derivatives of $f(z)$ for a fixed $z_0$ on the imaginary line?
That is, what is the growth rate of $\frac{d^n f(z)}{dz^n}(z_0)$ for a fixed $z_0$ as $n\to\infty$, where $z_0\in i \mathbb{R^+}$? In particular, what is the growth rate of $\frac{d^n f(z)}{dz^n}(i)$?