# Singular locus of dual hypersurfaces

Everything is over field $$\mathbb C$$. Let $$X$$ be a hypersurface of degree $$d$$ in $$\mathbb P^n$$. We know that if $$X$$ is smooth, then its dual $$X^\vee$$ is still a hypersurface in $$(\mathbb P^n)^\vee$$, but not necessarily smooth. I want to know that, for a general choose of $$X$$, can we compute the dimension of the singular locus $$X^\vee_{sing}$$? Where can I find a discussion of this?

For some reason, I believe in general it is of codimension $$1$$ In $$X^\vee$$, or it is empty. But I don’t know how to prove this.