These traveling-wave partial-derivative relations may be derived a bit more formally by means of the chain rule from calculus, which states that, for the composition of functions and , i.e.,
the derivative of the composition with respect to can be expressed according to the chain rule as
where denotes the derivative of with respect to .
To apply the chain rule to the spatial differentiation of traveling waves, define
Then the traveling-wave components can be written as and , and their partial derivatives with respect to become
and similarly for .