I'm struggling to proof that
$\int_{\mathbb{R}}P(z,t|x)\sum_{n=1}^{\infty}D^{(n)}(z)h^{(n)}(z)dz$
(with $D^{(n)}(z):=\frac{1}{n!}\lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\int_{\mathbb{R}}P(y,\Delta t|x)(y-z)^{n}dy$ and $P$ transition density of an Ito's process) is equal to
$\int_{\mathbb{R}}h(z)\sum_{n=1}^{\infty}(-\frac{\partial}{\partial z})^{n}[D^{(n)}(z)P(z,t|x)]dz$
I know that $\int_{\mathbb{R}}\sum_{n=1}^{\infty}D^{(n)}(z)h^{(n)}(z)dz\equiv \int_{\mathbb{R}}(\frac{\partial}{\partial z})^{n}[D(z)h(z)]dz=\int_{\mathbb{R}}\sum_{n=1}^{\infty}D^{(n)}(z)h(z)+h^{(n)}(z)D(z)$
but I'm having a hard time to understand how apply Fundamental Theorem of integral calculus in this case. I hope you can help me.
Thanks in advance!