I've had problems applying the Leibniz Rule.
The professor's slides say:
Consider the following value function \begin{equation} \nu(t) = \int_{t}^{\infty} e^{- \int_{t}^{s}r(z)dz} \pi(s) ds \end{equation} where $e^{- \int_{t}^{s}r(z)dz}$ is a discounting factor allowing to actualize of the flow of profits $\pi(s)$ and $r(z)$ is the discounting rate.
Then, deriving the previous expression with respect to $t$, the value function can be written in the form of the Hamilton-Jacobi-Bellman equation (using the Leibniz rule) :
\begin{equation} \dot{\nu}(t) + \pi(t) = r(t)\nu(t) \end{equation}
How do I get the last expression? From my math course, I understand the Leibniz rule as follows:
Consider, $y(t) = \int_{a(t)}^{b(t)}f(s,t)ds$, then \begin{equation} y'(t)= b'(t)f(b(t),t)-a'(t)f(a(t),t)+ \int_{a(t)}^{b(t)} \frac{\partial f(s,t)}{\partial t} ds \end{equation}