This is only part of the solution I got stuck on. I don't quite understand how can the instructor got from $\rm{d} (\log S_t) = \frac{dS_t}{S_t}$.
Thank you.
And also where did the negative sign coming from in the very last formula ? Shouldn't it be plus sign ? Thank you.
Remember what the derivative of a logarithm is: $$\frac{d}{dS_t}\log S_t = \frac{1}{S_t}$$
EDIT: As Eric S. points out, the second derivative of a logarithm is $$\frac{d^2}{dS_t^2}\log S_t=\frac{d}{dS_t}\frac{1}{S_t}=\frac{-1}{S_t^2}$$