How to differentiate $\int_{B(t)} f(x,t) dx$ with respect to $t$?

Question

$\int_{B(t)} f(x,t) dx$ is given and assume $x$ is an $n$ dimensional vector variable, $t$ is positive real variable and $f(x,t)$ is a sufficiently smooth function of both $x$ and $t$. Also $B(t)$ is a time varying region in the $d$ dimensional Euclidean space with sufficient smoothness conditions. Then what is the formula for differentiating the integral $\int_{0}^{t} f(x,t) dx$ with respect to $t$? Could anyone please explain?

Answer 1

Some clever guy has already thought about it: look here.

ADDENDUM

In case you are interested in differentiating with respect to $t$ the function $$ t\mapsto\int_0^t \dotsi \int_0^t f(x_1,\dots,x_n,t) \,dx_1\dots dx_n, $$ just notice that $$ \int_0^t \dotsi \int_0^t f(x_1,\dots,x_n,t) \,dx_1\dots dx_n = \int_0^t g(x_n,t) \,dx_n $$ where $$ g(x_n,t) = \int_0^t \dotsi \int_0^t f(x_1,\dots,x_n,t) \,dx_1\dots dx_{n-1}, $$ so Leibniz rule can be applied iteratively.