121616949-math.99 - 4.9 Implicit Differentiation 85 to...

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4.9 Implicit Differentiation 85 to assume that the desired derivative exists, but fortunately this is a safe assumption for most such problems. The example y = ln x involved an inverse function defined implicitly, but other func- tions can be defined implicitly, and sometimes a single equation can be used to implicitly define more than one function. Here’s a familiar example. The equation r 2 = x 2 + y 2 describes a circle of radius r . The circle is not a function y = f ( x ) because for some values of x there are two corresponding values of y . If we want to work with a function, we can break the circle into two pieces, the upper and lower semicircles, each of which is a function. Let’s call these y = U ( x ) and y = L ( x ); in fact this is a fairly simple example, and it’s possible to give explicit expressions for these: U ( x ) = r 2 - x 2 and L ( x ) = - r 2 - x 2 . But it’s somewhat easier, and quite useful, to view both functions as given implicitly by r 2 = x 2 + y 2 : both r 2 = x 2 + U ( x ) 2 and r 2 = x 2 + L ( x ) 2
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