Let $A$ be a symmetric, positive definite $p\times p$ matrix, and let $f(A)$ be its Cholesky factor. That is, $f(A)$ is a lower triangular $p\times p$ matrix such that $A = f(A) f(A)^{\top}$. I am wondering if the derivative $$ \frac{\mathrm{d}\operatorname{vech}\left(f(A)\right)}{\mathrm{d}\operatorname{vech}\left(A\right)} $$ is known, where $\operatorname{vech}$ is the half-vectorization function.

(I am conjecturing that it is something like $L^{\top} \left(f(A)\otimes A^{-1}\right) L$, where $L$ is the elimination matrix, but I would need a reference or proof anyway.)