To what extent $\frac{d}{dx} \int_{-\infty}^{\infty} f(t,x)\, dt = \int_{-\infty}^{\infty}\frac{d}{dx}f(t,x)\, dt$

Question

Assume $\int_{-\infty}^{\infty} f(t,x)\, dt$ exists for every $x \in {\mathbb R}^n$. Then, to what extent, the following equation holds?

$$\frac{d}{dx} \int_{-\infty}^{\infty} f(t,x)\, dt = \int_{-\infty}^{\infty}\frac{d}{dx}f(t,x) \,dt.$$

Answer 1

I think this answers your question for the finite case in Theorems 1 and 2. Extending it to the unbounded case is achieved in Theorems 3-5.

Without retyping all the proofs, $f$ and $\frac{d}{dx}f$ need to be continuous, $\int_{-\infty}^{\infty} f$ and $\int_{-\infty}^{\infty} \frac{d}{dx}f$ need to be uniformly convergent.

Answer 2

With the Lebesgue integral we have this theorem (Integración de funciones de varias variables):

Let be $(X,\mu)$ a measure space, $J\subset\Bbb R$ an interval.

Exists $g$ integrable s.t. $|f(x,t)|\le g(x)$ for all $t\in J$ and almost all $x\in X$.

Exists $\partial_t f(x,t)$ for all $t\in J$ and almost all $x\in X$.

Exists $h$ integrable s.t. $|\partial_t f(x,t)|\le h(x)$ for all $t\in J$ and almost all $x\in X$.

Then $$F(t) = \int_X f(x,t)\,d\mu(x)$$ is derivable and $$F'(t) = \int_X \partial_t f(x,t)\,d\mu(x).$$