Let $f(x)$ be an integrable (not necessarily differentiable) function on $[a,\infty)$. Let $g(x,y)$ be differentiable on $[a,\infty)\times [c,d)$. I can also assume that the partial derivative of $g(x,y)$ with respect to $y$ is continuous.
Define $$h(y) = \int_a^\infty g(x,y)\,f(x)\,dx.$$
When calculating the differentiation $h'(y)$ for $y\in[c,d)$, can I always put the differentiation under the integral sign?
(If not, let me know if it will work with any additional "reasonable" assumptions.)