This is a follow-up on differentiating the mgf to find moments.
Let $g(t,x)$ be a smooth function and $X$ be a random variable. Set $f(t) = E[g(t,X)]$. If $g_t(t,X)$ is dominated by an $L^1$ random variable, then one can show $f$ is differentiable.
Is there an example of a random variable $X$ and smooth $g(t,x)$ such that $f(t) = E[g(t,X)]$ is finite and smooth for all $t$ but $E[g_t(t,X)] = \infty$?
Let $X$ be a standard normal, and let $$g(t,x) = \exp(-(t e^{x^2}-1)^2)$$ $$g_t(t,x) = (2-2te^{x^2})\exp(-(t e^{x^2}-1)^2+x^2)$$ $$g_t(0,x) = (2/e)e^{x^2}$$
Then $g(t,X)$ has finite expectation, since $g(t,x)\le 1$, but $g_t(0,X)$ has infinite expectation.
Here is the contour plot of $g$, with white regions near 1 and red regions near 0.