The Chain Rule in a PDE Involving a Natural Logarithm (Larson 9e 13.3.28)

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(x² +y²)^1/2 in terms of three functions:

• Outermost function: f(u) = ln(u), with u = (x² +y²)^1/2
• Next function (inside the natural logarithm): g(v) = v^1/2, with v = x² +y² 
• Innermost function (inside the square root): h(x, y) = x² +y² 

Note that, in taking the partial derivative with respect to x, we only differentiate x itself in the innermost function, where, of course, x² +y² becomes 2x. 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 x² +y². 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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