How to differentiate such a integral with respect to $\theta$
$$0 =\frac{d}{d\theta}\int_{0}^{\infty} a(\theta)h(x)e^{T(x)^T \theta} dx$$
where $a(\theta), h(x)$ are given functions $T(x)^T$ is transpose of $T(x)$, $\theta \in R^k$
I want to solve this equation to get the expectation of $T(x)$
I am kind of new to vector or matrix calculas, so any help or hint is welcome.
by the Leibniz integral rule: $$\begin{align}\phantom{=}&\frac{d}{d\theta}\left[\int\limits_0^\infty a(\theta)h(x)\exp(T(x)^T\theta)\,dx\right]\\ =&\int\limits_0^\infty\frac{\partial}{\partial\theta}\left[a(\theta)\exp(T(x)^T\theta)\right]h(x)\,dx\end{align}$$ now use chain rule for this derivative: $$\frac{\partial}{\partial\theta}\left[a(\theta)\exp(T(x)^T\theta)\right]=a'(\theta)\exp(T(x)^T\theta)+a(\theta)T(x)^T\exp(T(x)^T\theta)$$ In terms of simplifying I am not sure as I do not know the form of $a$ but if $a\in\mathbb{R}^k$ then you should instead say: $$\exp(T(x)^T\theta)a'(\theta)+T(x)^Ta(\theta)\exp(T(x)^T\theta)$$ I hope this helps! If anyone spots a mistake please let me know