Use of the Chain Rule

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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.,

$\displaystyle y(x) = f(g(x)),$

the derivative of the composition with respect to

can be expressed according to the chain rule as

$\displaystyle y'(x) = f^\prime(g(x))g^\prime(x),$

where $f^\prime(x)$ denotes the derivative of

with respect to

To apply the chain rule to the spatial differentiation of traveling waves, define

$\begin{eqnarray*} g_r(t,x) &=& t - \frac{x}{c}\\ [10pt] g_l(t,x) &=& t + \frac{x}{c}. \end{eqnarray*}$

Then the traveling-wave components can be written as and , and their partial derivatives with respect to become

$\begin{eqnarray*} y'_r\;\isdef \; \frac{\partial}{\partial x} y_r\left[g_r(t,x)\... ...t \left(-\frac{1}{c}\right) \;\isdef \; -\frac{1}{c}{\dot y}_r, \end{eqnarray*}$

and similarly for .