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- Differentiation_under_the_integral_sign abstract "Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows:Theorem. Let f(x, t) be a function such that both f(x, t) and its partial derivative fx(x, t) are continuous in t and x in some region of the (x, t)-plane, including a(x) ≤ t ≤ b(x), x0 ≤ x ≤ x1. Also suppose that the functions a(x) and b(x) are both continuous and both have continuous derivatives for x0 ≤ x ≤ x1. Then for x0 ≤ x ≤ x1:This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The [second] fundamental theorem of calculus is just a particular case of the above formula, for a(x) = a, a constant, b(x) = x and f(x, t) = f(t).If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:ItDx = DxIt,where Dx is the partial derivative with respect to x and It is the integral operator with respect to t 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.The following three basic theorems on the interchange of limits are essentially equivalent: the interchange of a derivative and an integral (differentiation under the integral sign; i.e., Leibniz integral rule) the change of order of partial derivatives the change of order of integration (integration under the integral sign; i.e., Fubini's theorem)".
- Differentiation_under_the_integral_sign wikiPageID "1065076".
- Differentiation_under_the_integral_sign wikiPageRevisionID "604814381".
- Differentiation_under_the_integral_sign hasPhotoCollection Differentiation_under_the_integral_sign.
- Differentiation_under_the_integral_sign subject Category:Differential_calculus.
- Differentiation_under_the_integral_sign subject Category:Integral_calculus.
- Differentiation_under_the_integral_sign comment "Differentiation under the integral sign is a useful operation in calculus. Formally it can be stated as follows:Theorem. Let f(x, t) be a function such that both f(x, t) and its partial derivative fx(x, t) are continuous in t and x in some region of the (x, t)-plane, including a(x) ≤ t ≤ b(x), x0 ≤ x ≤ x1. Also suppose that the functions a(x) and b(x) are both continuous and both have continuous derivatives for x0 ≤ x ≤ x1.".
- Differentiation_under_the_integral_sign label "Differentiation under the integral sign".
- Differentiation_under_the_integral_sign label "Integral paramétrica".
- Differentiation_under_the_integral_sign label "Intégrale paramétrique".
- Differentiation_under_the_integral_sign label "Parameterintegral".
- Differentiation_under_the_integral_sign label "积分符号内取微分".
- Differentiation_under_the_integral_sign sameAs Parameterintegral.
- Differentiation_under_the_integral_sign sameAs Intégrale_paramétrique.
- Differentiation_under_the_integral_sign sameAs Integral_paramétrica.
- Differentiation_under_the_integral_sign sameAs m.042xm_.
- Differentiation_under_the_integral_sign sameAs Q286461.
- Differentiation_under_the_integral_sign sameAs Q286461.
- Differentiation_under_the_integral_sign wasDerivedFrom Differentiation_under_the_integral_sign?oldid=604814381.
- Differentiation_under_the_integral_sign isPrimaryTopicOf Differentiation_under_the_integral_sign.