In the context of applying the chain rule for tensors, the $$\textrm{prod}$$ operator is a conceptual tool used to multiply matrices of partial derivatives after performing necessary structural operations, such as transpositions and swapping input positions. For vectors, this operation simplifies directly to standard matrix–matrix multiplication. The operator encapsulates all the notational overhead and dimensional adjustments required when multiplying derivatives of higher-dimensional tensors.

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The chain rule from calculus can be generalized to compute the derivative of a tensor $$\mathsf{Z}$$ with respect to a tensor $$\mathsf{X}$$ via an intermediate tensor $$\mathsf{Y}$$, where all three tensors have arbitrary shapes. Assuming functions $$\mathsf{Y}=f(\mathsf{X})$$ and $$\mathsf{Z}=g(\mathsf{Y})$$, the chain rule is expressed as:
$$ \frac{\partial \mathsf{Z}}{\partial \mathsf{X}} = \textrm{prod}\left(\frac{\partial \mathsf{Z}}{\partial \mathsf{Y}}, \frac{\partial \mathsf{Y}}{\partial \mathsf{X}}\right) $$

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