Hyperosculating point count

Question

This is exercise IV.4.6(b) of Harshorne. Let $X$ be a (smooth) curve of genus $g$ embedded as a curve of degree $d$ inside $P^n$, $n\geq 3$, and not contained in any hyperplane. Then I would like to calculate the number of hyperosculating points. I suppose this is related to the map $X\rightarrow X^*$, the map to its dual curve. However, to apply Rimann-Hurwitz, I need to know the genus of the dual curve. Also, I am unclear of the relation between the ramification locus and the number of hyperosculating points.