Is there any way of doing the below but by avoiding $d^3f\over dx^3$?
If for a row vector $x$ e.g. size (3x1):
$f=f(x)=scalar$
$g={df\over dx}$ i.e. the gradient
$H={d^2f\over dx^2}$ i.e. the Hessian
$M=fH$
Then the derivate of $M$ w.r.t. $x$ is given by
$$dM={dM\over df}df+{dM\over dH}dH$$ $$dM={H}df+{f}dH$$ $$dM={H}(gdx)+{f}{dH\over dx}dx$$ $${dM\over dx}=Hg+{f}{dH\over dx}$$ $${dM\over dx}=Hg+{f}{df^3\over dx^3}$$