Is it possible ? move diff outside in integral

Question

I have a question. When we have following integral,

$$ \int _{\ }^{\ }a\ \cdot \ \frac{db}{dt}dx $$

Can we move differential to outside of integral like this ? If possible could you tell me the rule name ?

$$ \frac{d}{dt}\int _{\ }^{\ }a\ \cdot \ b\ dx $$

a is some sort of constant or function. and b is vector(i,j,k)

Thanks you..

Answer 1

Robert Lee commented the correct result. The rule is called the Leibniz integral rule. So long as $b(x,t)$ is smooth in the region being integrated, then it is generally true.