### 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.,

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 .

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