Let $f :X \rightarrow \mathbb{R}$ be a bounded and continuously differentiable function on $X =(a ,b) \subset \mathbb{R}$. $P$ is a Markov kernel on $X$. Can we say something about the derivative of the function $M(x) : =\int _{X}f(x^{ \prime })P(x ,dx^{ \prime })$? Is it well defined? exist (under suitable assumptions on $P$)?
One example: If $P$ is a Markov kernel of a AR(1) process $x^{ \prime } =x +Z$ for some random variable $Z$ with law $\pi $. Then
$M(x) =\int f(x +Z)\pi (dZ)$, so $M^{ \prime }(x) =\int f^{ \prime }(x +Z)\pi (dZ)$.