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3.7 Implicit Differentiation

3.7 Implicit Differentiation - (3.7 Implicit Dierentiation...

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(3.7) Implicit Differentiation. Example. Find d dx y 3 if y = cos x . We get - 3 cos 2 x sin x by the chain rule. What if y = x 4 + 5 x ? Then we get 3( x 4 + 5 x ) 2 (4 x 3 + 5). In each case, we get 3 y 2 dy dx . This makes sense by the chain rule: Say w = y 3 . Then dw dx = dw dy dy dx = 3 y 2 dy dx . Example. Now try 1. x 2 y 3 2. x 2 /y 4 3. sin y Implicit functions. We can specify a function y explicitly, by stating a formula for it. Say y = 25 - x 2 . (‘Explicit’ means ‘fully revealed’.) But we can specify the same function implicitly: x 2 + y 2 = 25. We understand that we can solve for y to obtain the explicit formula. (This also gives y = - 25 - x 2 implicitly.) ‘Implicit’ means ‘capable of being understood from something else though unexpressed’. Another example: The function given implicitly by y 3 +3 y = 2 x can be given explicitly by solving for y . However, this is not easy in this case. It turns out y = 3 x + x 2 + 1 + 3 x - x 2 + 1 . It often happens that it is difficult or impossible to deal with a function explicitly, and instead, we must deal with it implicitly. (This is particularly true in the subject known
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