# Leibniz Integral Rule

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.

### Other articles related to "leibniz integral rule, integral, integrals, rule":

Leibniz Integral Rule - Proofs - Three-dimensional, Time-dependent Form
... See also Differentiation under the integral sign in higher dimensions At time t the surface Σ in Figure 1 contains a set of points arranged about a ... integration are then independent of time, so   where the limits of integration confining the integral to the region Σ no longer are time dependent so differentiation passes through the ... that Stokes theorem allows the surface integral of the curl over Σ to be made a line integral over ∂Σ   The sign of the line integral is based on ...
Differentiation Under The Integral Sign - Higher Dimensions
... The Leibniz integral rule can be extended to multidimensional integrals ... In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem where is a scalar function, and denote a time-varying ... The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior ...
Differentiation Under The Integral Sign
... Differentiation under the integral sign is a useful operation in calculus ... for This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus ... where is the partial derivative with respect to and is the integral operator with respect to over a fixed interval ...

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