Chain Rule Notation

The chain rule has several equivalent notations, each useful in different contexts. Let $\mathbf{h}(\mathbf{x}) = \mathbf{g}(\mathbf{f}(\mathbf{x}))$ with $\mathbf{u} = \mathbf{f}(\mathbf{x})$:

Jacobian notation: $J\mathbf{h}(\mathbf{x}) = J\mathbf{g}(\mathbf{u}) \cdot J\mathbf{f}(\mathbf{x})$ — clean and coordinate-free.

Leibniz notation: $\frac{\partial h_i}{\partial x_j} = \sum_{k=1}^K \frac{\partial g_i}{\partial u_k} \frac{\partial f_k}{\partial x_j}$ — explicit coordinate formula, useful for computation.

Index notation: $(J\mathbf{h})_{ij} = \sum_k (J\mathbf{g})_{ik}(J\mathbf{f})_{kj}$ — this is exactly matrix multiplication.

The Leibniz form is often encountered in physics and engineering. The index form reveals the chain rule as matrix multiplication — the sum over $k$ is the dot product of row $i$ of $J\mathbf{g}$ with column $j$ of $J\mathbf{f}$.