Abstract
An efficient coordinate-free notation is elucidated for differentiating matrix expressions and other functions between higher-dimensional vector spaces. This method of differentiation is known, but not explained well, in the literature. Teaching it early in the curriculum would avoid the tedium of element-wise differentiation and provide a better footing for understanding more advanced applications of calculus. Additionally, it is shown to lead naturally to tensor products, a topic previously considered too difficult to motivate quickly in elementary ways.
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