$$ P_s(a)=\int_{-\infty}^{\infty}du\int_{-\infty}^{a-u}dv\,p_{x,y}(u,v) \\[15pt] p_s(a) = \frac{d}{da}P_s(a) = \int_{-\infty}^\infty du\,p_{x,y}(u,a-u) $$
Please help to understand how this follows or suggest the reading or keywords to look up. Precisely, what makes/allows applying differentiation to the second (nested) integral?
Source (page 3 in the pdf, or page 49 in the book): https://ocw.mit.edu/courses/physics/8-044-statistical-physics-i-spring-2013/readings-notes-slides/MIT8_044S13_ProbabilityCh4.pdf