Leibniz integral rule for improper integral with variable limit

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Given an integral of the form $F(x)=\int_{t_0}^x f(x,t)dt$, where the function $f(x,t)$ is infinite at $t=x$, what are the minimal assumptions on $f(x,t)$ for the Leibniz integral rule to apply? I need to derive $F'(x)$.