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Concept
Leibniz integral rule
Summary
In calculus, the Leibniz integral rule for differentiation under the integral sign states that for an integral of the form
where and the integrands are functions dependent on the derivative of this integral is expressible as
where the partial derivative indicates that inside the integral, only the variation of with is considered in taking the derivative. It is named after Gottfried Leibniz.
In the special case where the functions and are constants and with values that do not depend on this simplifies to:
If is constant and , which is another common situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes:
This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and 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.
The right hand side may also be written using Lagrange's notation as:
Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous. This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where is constant, and does not depend on
If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:
where is the partial derivative with respect to and is the integral operator with respect to over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.
Official source
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