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An efficient coordinate-free notation is elucidated for differentiatingmatrix expressions and other functions between higher-dimensional vectorspaces. This method of differentiation is known, but not explained well, in theliterature. Teaching it early in the curriculum would avoid the tedium ofelement-wise differentiation and provide a better footing for understandingmore advanced applications of calculus. Additionally, it is shown to leadnaturally to tensor products, a topic previously considered too difficult tomotivate quickly in elementary ways.

Differential Calculus, Tensor Products and the Importance of Notation