I am currently asking myself whether I did everything correctly when differentiating an integral with respect to its upper limit. I need the solution of this differentiation for further calculations in R and want to make sure the results I get out of my routine are not wrong.
Let $F(0)=G(0)=0$, Is it correct that
$$\dfrac{d}{dt}\int_{x=0}^{x=t}\int_{u=0}^{u=x}\,\mathrm dF(u)\,\mathrm dG(x) = G'(t)\int_{u=0}^{u=t}\,\mathrm dF(u) $$
provide $G'(t)$ exists?
Many thanks in advance.
Yes indeed it is true. It can be broken down further as shown. $$\dfrac{d}{dt}\int_{x=0}^{x=t}\int_{u=0}^{u=x}\,\mathrm dF(u)\,\mathrm dG(x) =\dfrac{d}{dt}\int_{0}^{t}\int_{u=0}^{u=x}\,\mathrm dF(u) \,G'(x)\,\mathrm dx = G'(t)\int_{u=0}^{u=t}\,\mathrm dF(u)\\= G'(t)\int_{u=0}^{u=t}\,\mathrm dF(u)$$