Leibniz integral rule

The Leibniz integral rule gives the derivative of a parameter-dependent definite integral by accounting for changes in its integrand and integration limits.

Leibniz rule with fixed limits: A differentiation formula that moves a derivative inside an integral over constant bounds, under suitable regularity conditions. This is the rule’s simpler form when only the integrand depends on the parameter.
Definite integral: A number representing accumulated signed area or another limit of sums over an interval. The rule begins with a definite integral whose value varies with a parameter.
Feynman's integration trick: A method for evaluating an integral by introducing a parameter, differentiating with respect to it, and integrating back. It uses the Leibniz rule to replace a hard integral with a simpler parameter derivative.
Gottfried Wilhelm Leibniz: A German polymath who independently developed calculus and introduced influential notation for it. The rule bears his name and uses ideas expressed in his differential notation.
Leibniz rule with variable limits: A differentiation formula that includes endpoint terms when one or both integration limits depend on a parameter. This form makes the boundary contributions explicit.
Partial derivative: A derivative of a multivariable function with respect to one variable while holding the others fixed. The integrand’s parameter dependence is differentiated while its integration variable is held fixed.
Score function identity: An identity expressing the derivative of an expectation as an expectation involving the log-density derivative. It is a probabilistic application of differentiating parameter-dependent integrals.
Isaac Newton: An English mathematician and physicist who independently developed calculus using the language of fluxions. Newton’s fluxional approach offers a historical contrast to Leibniz’s differential notation.
Fundamental theorem of calculus: A pair of theorems connecting differentiation and integration. It supplies the endpoint terms when a limit of integration moves.
Parameter-dependent integral: An integral whose integrand, domain, or limits depend on one or more parameters. This is the general object to which the rule applies.

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