  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.
