Here’s a problem (13.3.28, p. 914) from Larson et al.’s 9^{th} edition of Calculus:

This problem is notable because it illustrates the possible complexity of the chain rule when applied to a logarithm that is being differentiated. Think of ln(*x*^{2} +*y*^{2})^{1/2} in terms of three functions:

- Outermost function:
*f*(*u*) = ln(*u*), with*u*= (*x*^{2}+*y*^{2})^{1/2} - Next function (inside the natural logarithm):
*g*(*v*) =*v*^{1/2}, with*v*=*x*^{2}+*y*^{2} - Innermost function (inside the square root):
*h*(*x*,*y*) =*x*^{2}+*y*^{2}

Note that, in taking the partial derivative with respect to *x*, we only differentiate *x* itself in the innermost function, where, of course, *x*^{2} +*y*^{2} becomes 2*x. *If the natural logarithm had not been raised to a non-1 power, then, of course, there would only have been the outer function of ln (*u*) itself and the inner function of *x*^{2} +*y*^{2}. The third function arose because the natural logarithm in this problem is raised to a power other than 1 and therefore has to be accommodated in the chain rule.

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