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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Derivatives