Say, $f:\mathbb R^n\to\mathbb R^k$ and $g:\mathbb R^k\to\mathbb R$ are both $C^2$. I'd like to express the Hessian matrix of $g\circ f$
$$\left( \frac{\partial^2(g\circ f)}{\partial x_i \partial x_j} (x) \right)_{i,j\in\{1,\ldots, n\}}$$
in terms of partial derivatives of $g$ and $f$.
I know that $$(g\circ f)''(x)[v,w]=f''(g(x))\Big[g'(x)[v], g'(x)[w]\Big]+f'(g(x))\circ g''(x)[v,w]$$
yet I have problems writing it out in terms of partial derivatives.
I think your formula switched $f$ and $g$, I found:
$(g\circ f)''(x)[u,v]=g''(f(x)[f'(x)[u], f'(x)[v]]+g'(f(x))\circ f''(x)[u,v]$.
Write $f=(f^1,...,f^k)$ and plugging in $e_i=u, e_j=v$ gives:
$\frac{\partial (g\circ f)}{\partial x_i\partial x_j}=\frac{\partial f^h}{\partial x_i}\frac{\partial f^\ell}{\partial x_j}\frac{\partial^2 g}{\partial x_h\partial x_{\ell}}+\frac{\partial g}{\partial x_m }\frac{\partial^2 f^m}{\partial x_i\partial x_j}$
Where term's are summed over $1\le h,\ell, m\le k$.