It's nice to read blog posts on calculus and linear algebra, but I'm affraid that the author simply skipped some basic lectures on the topic if he was oblivious to the fact that the Hessian reflects the second derivative of a multi-variate function. That's the whole reason why all newton and quasi-newton methods rely on the Hessian (and its approximations) to find local maximum/minimum of a function.

Heck, IIRC Apostol's 2nd volume dedicates a whole chapter on how to derive the Hessian, including from directional derivatives.