### Use of the Chain Rule

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.*,

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 .

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Traveling-Wave Partial Derivatives