# Differentiation of value function in perpetual american option

I am trying to solve the perpetual American option problem. Currently I'm following this (slide 9). The stock price is modelled as Ito's process.

$$dS_t = (\mu-D_0)S_tdt\ +\ \sigma S_tdW_t$$

where $$D_0$$ is the dividend. Basically the problem tries to find optimal stopping time to exercise the option i.e. we try to optimize the value function $$V(S)$$ w.r.t stopping time $$\tau$$.

$$V(S) = sup_{\tau \in T}\ \mathbb{E}[e^{-r\tau}max(S_\tau -K,\ 0)]$$

I'm stuck in deriving the partial derivative of $$V(S)$$ w.r.t. time.

In my opinion $$V(S)$$ can be written as follows (where we integrate over all the values of time considering the probability density function of time $$u$$ being the stopping time $$\tau$$)

$$V(S) =sup_{\tau \in T}\ \int_0^\infty\ e^{-ru}max(S_u-K,\ 0)\ \mathbb{P}(\tau = u)\ du$$

According to the reference (slide 9), the partial derivative of $$V(S)$$ w.r.t. time comes out to be $$-rV(S)$$ which is not very clear to me. So please help me out with in-depth proof, if possible.

Thanks :)