I have trouble deriving the following equation with respect to t (that is, $\frac{d}{dt}$): $$ V_{\nu} (t) = \int_{t}^{\infty } e^{-\int_{t}^{s} r(s') ds'} \pi_{\nu}(s) ds$$ (I know basic derivative rules but am not sure how to handle the Euler-function part of the equation). Would be very happy about help! Thank you in advance!
The function is an economic function describing the profit discounted over time (the integral expression). r is the rental rate of capital and s is just some time index other than t. s' is not any derivative and is just independent from s!
Let $V_\nu(t)=\int_t^\infty e^{-\int_t^s r(s')\,ds'}\pi_\nu(s)\,ds$. Then, we have from Leibniz's Rule
$$\begin{align} \frac{d}{dt}V_\nu(t)&=\frac{d}{dt}\int_t^\infty e^{-\int_t^s r(s')\,ds'}\pi_\nu(s)\,ds\\\\ &=-\left.\left(e^{-\int_t^s r(s')\,ds'}\pi_\nu(s)\right)\right|_{s=t}+\int_t^\infty \frac{\partial}{\partial t}\left(e^{-\int_t^s r(s')\,ds'}\,\pi_\nu(s)\right)\,ds\\\\ &=\pi_\nu(t)+\int_t^\infty \left(r(t)e^{-\int_t^s r(s')\,ds'}\right)\pi_\nu(s)\,ds\\\\ &=\pi_\nu(t)+r(t)V_\nu(t) \end{align}$$