I am reading a proof of Feynman-Kac theorem as done here, where I do not follow one step. Specifically, after the author derived: $$dY_s = u_x(t-s, W_s)e^{-R(s)}dW_s$$ they seem to have concluded $$Y_t =u(0,W_t)e^{-R(t)}$$ in the next line. But the partial derivative in $x$ magically seemed to have disappeared.
Can anyone point out if this is a mistake or this is how it is supposed to be?
It's a lot simpler than you thought. The definition of $Y_s$ is given in the beginning of the proof. He doesn't use the expression for $dY_s$ in any way in writing down $Y_t$.
At the beginning he defines $Y_s=e^{-R(s)}u(t-s,W_s)$. Letting $s=t$ gives $Y_t=e^{-R(t)}u(0,W_t)$.