Here is something which I am skeptical that it might be wrong:
$V$ is a two-variable $C^{\infty}$ function such that $V(t,w) = e^{-rt} F(w)$, for some one-variable $C^{\infty}$ function $F$. Let $z = V_w (t,w)$ and define a function $J$ such that $$ J(t,z) = V(t,w) - wz.$$
Then it asserts that $J_t = V_t$.
(But I get the following: $$J(t,z) = V(t,w) - w V_w (t,w) = e^{-rt} ( F(w) - wF'(w) ),$$ hence $$J_t= -r e^{-rt} ( F(w) - wF'(w) ) \neq V_t.)$$
Can anyone tell me which is right?
We use the definition of the partial derivative to show that this is true. Explanations for OP's issues are suggested in comments.
$$\begin{align} J_t(t,z) &= \lim_{\Delta t\rightarrow 0}\frac{J(t+\Delta t,w)-J(t,w)}{\Delta t}\\ &=\lim_{\Delta t\rightarrow 0}\frac{(V(t+\Delta t,w)-wz)-(V(t,z)-wz)}{\Delta t}\\ &=\lim_{\Delta t\rightarrow 0}\frac{V(t+\Delta t,w)-V(t,z)}{\Delta t}\\ &=V_t(t,w) \end{align} $$