Find expression for $\frac{d}{dn}\overline{s}_{\overline{n|}}$

Question

I got a question from book Mathematics of Investment and Credit

question 2.3.25 b) Find expression for: $$\frac{d}{dn}\overline{s}_{\overline{n|}}$$

I got the solution, but there is one part that I do not clear: $$\frac{d}{dn} \int_o^n(1+i)^{n-t}dt =1\cdot(1+i)^{n-n}-0\cdot(1+i)^{n-0}+\int_o^n \frac{d}{dn} e^{\delta(n-t)} $$

Can any one give me some hints why this equation make sense? Thanks

Answer 1

This is via a difficult to prove (and remember) formula for differentiating under the integral sign. It is a direct application of the theorem presented at the top of the Wikipedia page that I linked.