4.6 Differentiation under the Symbol of Integration (Leibniz Rule)

See “Handbook of Mathematics” by Bronshtein p457 differentiation under the symbol of integration.

\[\frac{d}{dy}\int_{a}^{b}{f(x,y)dx} = \int_{a}^{b}{\frac{\partial f(x,y)}{\partial y} dx}\]

More generally (when a, b is dependent on y)

\[\frac{d}{dy} \int_{\alpha(y)}^{\beta(y)}{f(x,y)dx} = \int_{\alpha(y)}^{\beta(y)}{\frac{\partial f(x,y)}{\partial y} dx} + \beta'(y)f(\beta(y),y) - \alpha'(y)f(\alpha(y),y)\]

Because time t is not dependent on the parameters, fortunately this is not the case for us!

이것은 미분방정식으로 표현된 구조모형(structural equation)에서 목적함수에 필요한 편미분값(예를 들어, \(\frac{\partial F}{\partial \eta_{i}}\)) 들을 구하는 데 매우 유용하게 사용된다.

Ref) 김정수 등. 미적분학. p 213-214. 이우출판사. 1987