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Math Help - differentiable

  1. #1
    Bop
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    differentiable

    Hi!

    I have this funtion: {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}} and if (x,y)=(0,0) \rightarrow f(x,y)=0

    so if I want to demonstrate it is differentiable in (x,y)=(0,0), here is what I do:

    1) I check if f(x,y) is continuous in (x,y)=(0,0).
    2) I find partial derivatives.
    3) I check if partial derivatives are continuous in (0,0).

    In case steps 1) and 3) are affirmatives, the function is differentiable in (x,y)=(0,0).

    Too wrong?

    So, in this case:

    1)It is continuous.

    2) I calculate partial derivate respect x: \dfrac{\partial f}{\partial x}=\displaystyle\frac{x^{4}+3x^{4}y^{2}}{{(x^{2}+y  ^{2})}^{2}}

    When I calculate the limit of partial derivate respect x in (0,0) and it doesn't exit, so partial derivates are not continuos in (0,0) so f(x,y) is not differentiable,
    Right?

    Thank you very much! Great forum!
    Last edited by Bop; January 17th 2010 at 04:59 AM. Reason: correction
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  2. #2
    MHF Contributor Calculus26's Avatar
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    Why is continuous at (0,0) ? what is f(0,0) ?
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  3. #3
    Bop
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    I forgot to define it in (0,0), thank you!
    Last edited by Bop; January 16th 2010 at 07:50 AM.
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  4. #4
    Bop
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    I have corrected the solution, can you have a look?

    Thank you!
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