Multiple Outputs

The general case $\mathbf{h} = \mathbf{g}\circ\mathbf{f}$ with vector-valued $\mathbf{g}$ (multiple outputs) follows the same Jacobian multiplication formula. Each output component $h_i = g_i(\mathbf{f}(\mathbf{x}))$ satisfies the chain rule, and stacking all outputs gives the Jacobian form.

Concretely, for $\mathbf{f}: \mathbb{R}^n \to \mathbb{R}^k$ and $\mathbf{g}: \mathbb{R}^k \to \mathbb{R}^m$: the $(i,j)$ entry of $J\mathbf{h} \in \mathbb{R}^{m\times n}$ is $[J\mathbf{h}]_{ij} = \frac{\partial h_i}{\partial x_j} = \sum_{k=1}^K \frac{\partial g_i}{\partial u_k}\frac{\partial f_k}{\partial x_j} = [J\mathbf{g}]_{ik}[J\mathbf{f}]_{kj}$ summed over $k$ — matrix multiplication entry by entry.

This is the most general and powerful form. It subsumes all special cases: scalar input ($n=1$, tangent vector), scalar output ($m=1$, gradient), and everything in between.