Let's suppose that $$\frac{d}{dx}\int_a^b f(x,t)\,dt=\int_a^b\frac{\partial}{\partial x}f(x,t)\,dt$$ for all $a,b\in\mathbb{R}$.
Can we extend this result to:
$$\frac{d}{dx}\int_{-\infty}^\infty f(x,t)\,dt=\int_{-\infty}^\infty\frac{\partial}{\partial x}f(x,t)\,dt$$?
Thanks for any help.
Reason for me asking this is because there is a nice result that says differentiation under the integral is ok, provided $f$, and $f_x$ are continuous, and the domain of integration must be compact. This is much easier than finding a dominating function in case we want to use Lebesgue's Dominated Convergence Theorem.
Eventually, I am hoping to use this to solve my other question: How do we justify differentiating under the integral for Poisson Integral