# Differentiating under integral sign, capital accumulation

I am reading a paper, the author takes the derivative of the following equation with respect to $t$: $$q(t) = \int_t^\infty e^{-r(s-t)}c(s)e^{-\delta(s-t)}ds$$

• $q$ price of capital goods
• $r$ discount rate
• $c$ cost of goods
• $\delta$ rate of replacement
• $t$ time of acquisition of goods
• $s$ time of supply of goods

I have the answer, but I was hoping someone would explain the intermediate points. Here, is the process similar to differentiating the functions with summation operators in discrete cases?

• It's just the Leibniz rule: en.wikipedia.org/wiki/Leibniz_integral_rule – Herr K. Feb 26 '16 at 19:29
• Can you please include in your question an expression that exemplifies the discrete case you have in mind? – Alecos Papadopoulos Feb 26 '16 at 19:38
• @Alecos, I mean the Riemann sum. – london Mar 1 '16 at 12:12
• Can anyone walk me through the solution? Thanks! – london Mar 1 '16 at 12:13