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Unformatted text preview: 3.7. HIGHER DERIVATIVES 345 and secondorder partials f xx = 2 f 2 x = 2 f xy = 2 f yx = 2 + 3 y 2 f yx = 2 f xy = 2 + 3 y 2 f yy = 2 f 2 y = 6 xy. It is clear that the game of successive differentiation can be taken further; in general a sufficiently smooth function of two ( resp . three) variables will have 2 r ( resp . 3 r ) partial derivatives of order r . Recall that a function is called continuously differentiable , or C 1 , if its (firstorder) partials exist and are continuous; Theorem 3.3.4 tells us that such functions are automatically differentiable. We shall extend this terminology to higher derivatives: a function is r times continuously differentiable or C r if all of its partial derivatives of order 1 , 2 ,...,r exist and are continuous. In practice, we shall seldom venture beyond the secondorder partials. The alert reader will have noticed that the two mixed partials of the function above are equal. This is no accident; the phenomenon was firstfunction above are equal....
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 Fall '08
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 Calculus, Derivative

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