How to Apply Leibnitz Rule When Integration Limits Depend on a Variable?

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The discussion centers on applying Leibnitz's rule for differentiation when the integration limits are functions of a variable. The original poster seeks guidance on finding the derivative of a function F(a,x) defined by an integral with variable limits dependent on 'a'. They note that traditional applications of Leibnitz's rule typically do not address this scenario. A key point raised is the need for clarification on how to handle cases where both limits of integration vary with the differentiation variable. The conversation highlights a gap in available resources covering this specific application of Leibnitz's rule.
Economist2008
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Does anybody know, how to find the derivative of the F with respect to a? As far as I know the Leibnitz rule is only applicable, when the integration limits do not depend a. But what happens, when one of the limits is a function of a?

F(a,x)=\int ^{c+h(a)}_{c} f[g(a,x)] dx

Thank you so much!
 
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Ups, the books I looked at didn't cover this case.
Thank you very much!
 

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