Consider the exponential family of distributions, whose density can be written as:
$$f_\theta(x)=h(x)\exp\left[\eta(\theta)^T T(x)-A(\theta)\right]$$
where $h: \mathbb{R}\to \mathbb{R}^+$, $\theta$ is a $k \times 1$ vector of parameters, $\eta: \mathbb{R}^k \to \mathbb{R}^L$ and $T:\mathbb{R}\to\mathbb{R}^L$.
Is the derivative $f^\prime_\theta(x)$ absolute integrable? That is, does the following integral always exists?
$$\int_{-\infty}^{+\infty} |f^\prime_\theta(x)| dx$$
Any proof or counter-example is welcomed.
EDIT: one can assume both $h$ and $T$ are continuous and differentiable every where.