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In general , how to show that the partial derivatives of a multivariable function exists without comupting it .

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That really depends on the circumstances. What else do you know about the function? –  fgp May 16 '13 at 21:36
    
I don't have a special situation but i am wondering how to show that a partial derivative exist , in general without need to compute it . –  user22323 May 16 '13 at 21:41
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Any polynomial function will have partial derivatives, for example. But once you take a fraction of polynomials, it's no longer necessary. More generally, anything you can get by a sum or product or composition of 'nice' functions like sin and cos and polynomials would have partial derivatives. –  user64480 May 16 '13 at 21:48
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Without a special situation, we can only refer to the definition of differentiability at a point, which involves the question of whether or not a limit exists, and thus in the general case, a computation. –  Jared May 16 '13 at 21:49
    
Sometimes, it is possible that you can form the difference quotients and show that the limit as $\epsilon \rightarrow 0$ exists, without explicitly computing said limit. –  Christopher A. Wong May 16 '13 at 22:07

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