A precise and reliable multivariable chain rule

The multivariable chain rule is often challenging to students because it is usually presented with ambiguities and other defects that hamper systematic and reliable application. A very simple formulation combines the derivation operators for functions and for expressions in a manner not found elsewhere due to common confusion between them. Some issues are rooted more deeply than others and are discussed in a broader perspective, starting with the function concept. The approach is illustrated using various applications including the transport equation, partial derivatives of a definite integral, and the distortionless (but not lossless) transmission line. This note is suitable for a lecture in any first-year course covering partial derivatives, as a complement to the other course material.