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Unformatted text preview: . Example: # 8 Chain Rule: Two Independent Variables Let w f x y = ( , ) , where f is a differentiable function of x and y . If x=g(s,t) and y=h(s,t) such that the first partials x s x t y s / , / , / , and y t / all exist, then w s / and w t / exist and are given by = + w s w x x s w y y s and = + w t w x x t w y y t . Example: #16 #20 Chain Rule: Implicit Differentiation If the equation F x y ( , ) = defines y implicitly as a differentiable function of x , then dy dx F x y F x y F x y x y y =  ( , ) ( , ) , ( , ) . If the equation F x y z ( , , ) = defines z implicitly as a differentiable function of x and y , then =  =  z x F x y z F x y z z y F x y z F x y z F x y z x z y z z ( , , ) ( , , ) , ( , , ) ( , , ) , ( , , ) . Example: #28 #34...
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 Spring '12
 PamelaSatterfield
 Approximation

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