The Leibniz integral rule supposedly requests $< \infty$ for the upper limit:
(...) an integral of the form $\int_{a(x)}^{b(x)}f(x,t)dt$, where $-\infty<a(x)$, $b(x)<\infty$, (...)
However, I can still do something like this:
$\displaystyle \int_{0}^{\infty} t^4e^{-xt} \text{dt} = \int_{0}^{\infty} \frac{\text{d}^4}{\text{d}x^4} e^{-xt} \text{dt} = \frac{\text{d}^4}{\text{d}x^4} \int_{0}^{\infty} e^{-xt} \text{dt} = \frac{\text{d}^4}{\text{d}x^4} x^{-1} = 24x^{-5}$
Why does this still work? Is this another rule?
