The Quotient Rule in Partial Derivatives: A Simple Trap to Avoid (Larson 9e 13.3.25)
Here’s a problem (13.3.25, p. 914) from Larson et al.’s 9th edition of Calculus:
This problem is notable because it illustrates a simple trap you can fall into when working with partial derivatives and the quotient rule. Consider that, when taking a partial derivative with respect to x in the problem above, any y term is a constant, and, therefore, the derivative of any y term is 0. This property of partial derivatives can be easy to forget when, after setting up the quotient rule, you might automatically set the derivative of y to 1. However, when we are differentiating with respect to x, then the derivative of any y term will be 0. If you go through the worked example above, you will see how, when we apply the quotient rule to differentiate in the x world, the derivatives of y terms are 0, and, when we differentiate in the y world, the derivatives of x terms are 0. This, in turn, affects the results yielded by the quotient rule.
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