A Question about Hessian of log function (general form)

Question

Sincerely hope to ask how to obtain the RHS?

Should I consider ln(10) among the process of d(log(x))/dx?

Thanks!

Answer 1

We can avoid computing the components of the Hessian individually by using the multivariable chain and product rules.

I'll assume that $f$ is differentiable and positive on an open set $\Omega \subset \mathbb R^N$. Let $g(x) = \log f(x)$ for all $x \in \Omega$. By the chain rule, \begin{equation} g'(x) = f(x)^{-1} f'(x). \end{equation}
(Note that $f'(x)$ is a $1 \times N$ row vector.)

It follows that \begin{align} \nabla g(x) &= g'(x)^T \\ &= f(x)^{-1} \nabla f(x). \end{align}

The Hessian of $g$ is given by $\nabla^2 g(x) = q'(x)$, where $q(x) = \nabla g(x)$. The product rule and the chain rule tell us that \begin{align} \nabla^2 g(x) &= q'(x) \\ &= -f(x)^{-2} \nabla f(x) \nabla f(x)^T + f(x)^{-1} \nabla^2 f(x). \end{align}