So here is all the slide says and I'm and trying to see the steps for derivation. Can someone please show me how this is derived? Essentially I'm trying to go from 1 to 2. Pi is the profit function.
If 1.)$$v(t)=max \pi(t,b)=\pi(t,B(t))$$
why does... 2.)$$v'(t)=d\pi/dt + (d\pi/db)B'(t)$$
Let me add that we are trying to use the envelope theorem here.
If I understand this right, you have a function $\pi(t, b)$, which is maximized for $b = B(t)$. If that is the case, straightforward application of the chain rule gives: $$ \frac{d v}{d t} = \frac{\partial \pi}{\partial t} + \frac{\partial \pi}{\partial b} \frac{d B}{d t} $$