Jacobian matrix of a composition of functions with chain rule Consider the differentiable functions $f:\mathbb{R}^{\ell}\to\mathbb{R}^{m}$, $g:\mathbb{R}^{m}\to\mathbb{R}^{n}$, and $h:\mathbb{R}^{n}\to\mathbb{R}^{o}$. Let $F:\mathbb{R}^{\ell}\to\mathbb{R}^{o}$ be the composition of these functions, i.e., $F=h\circ g\circ f$. We define the Jacobian matrix of $F$ evaluated at $\boldsymbol{x}\in\mathbb{R}^{\ell}$ as $J_{F}(\boldsymbol{x})\in\mathbb{R}^{o\times \ell}$, and similarly, we define the Jacobian matrices $J_{f}$, $J_{g}$, and $J_{h}$ for $f$, $g$, and $h$, respectively.
Using the chain rule, is it safe to say that $ J_F(x)=J_h(g\circ f(x))\cdot J_g(f(x))\cdot J_f(x) $ ?
If so, how can that be converted in matrix form ?
$$
J_F(x)=J_h(g\circ f(x))\cdot J_g(f(x))\cdot J_f(x)
$$
$$
J_F(x)=J_h(g\circ f(x))\cdot J_g(f(x))\cdot J_f(x)
$$