Leibniz Integral Rule
In Calculus, Leibniz's rule for differentiation under the integral sign, named after Gottfried Leibniz, tells us that if we have an integral of the form
then for the derivative of this integral is thus expressible
provided that and are both continuous over a region in the form
Thus under certain conditions, one may interchange the integral and partial differential operators. This important result is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's Integral Rule applies is essentially a question about the interchange of limits.
Read more about Leibniz Integral Rule: Formal Statement, Three-dimensional, Time-dependent Case
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