Given an exponential family distribution of the form
$$ f_X(x)=h(x)e^{\phi(\theta)^TT(x)-A(\theta)} $$
with natural parameter $\eta=\phi(\theta)$, sufficient statistic $T(x)$, and log partition function $A(\theta)$. We can absorb the $-A(\theta)$ term into the natural parameter vector by concatenating it:
$$ \phi'(\theta)=\left[\begin{array}{c}\phi(\theta)\\A(\theta)\end{array}\right],\quad T'(x)=\left[\begin{array}{c}T(x)\\-1\end{array}\right] $$
so that the log partition term disappears. We know that we can compute $\mathbb{E}[T'(x)]=\nabla_{\eta'}A'(\theta)$. Since $A'(\theta)=0$ this would mean that $\mathbb{E}[T'(x)]=[0,0,\cdots,0]^T$ for all distributions in the exponential family, which seems wrong. What is the problem here?