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Math Help - Check differentiability in function

  1. #1
    Bop
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    Check differentiability in function

    Hello, I can't find any way to prove if this funtion is or isn't differentiable in if (x,y)=(0,0):

    {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}} if (x,y) \neq(0,0)

    f(x,y)=0 if (x,y)=(0,0)

    Thank you!
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  2. #2
    No one in Particular VonNemo19's Avatar
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    Quote Originally Posted by Bop View Post
    Hello, I can't find any way to prove if this funtion is or isn't differentiable in if (x,y)=(0,0):

    {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}} if (x,y) \neq(0,0)

    f(x,y)=0 if (x,y)=(0,0)

    Thank you!
    Start by taking the derivative and seeing whether or not it is defined at (0,0).
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  3. #3
    Super Member General's Avatar
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    Quote Originally Posted by Bop View Post
    Hello, I can't find any way to prove if this funtion is or isn't differentiable in if (x,y)=(0,0):

    {f(x,y)=\displaystyle\frac{x^{3}}{x^{2}+y^{2}}} if (x,y) \neq(0,0)

    f(x,y)=0 if (x,y)=(0,0)

    Thank you!
    Use the following formulas to find f_x(0,0) and f_y(0,0):
    f_x(a,b)=\lim_{h\to0} \frac{f(a+h,b)-f(a,b)}{h}
    f_y(a,b)=\lim_{h\to0} \frac{f(a,b+h)-f(a,b)}{h}
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  4. #4
    Bop
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    Thanks for the answers!

    I have calculated derivative respect x: \dfrac{\partial f}{\partial x}=\displaystyle\frac{x^{4}+3x^{4}y^{2}}{{(x^{2}+y  ^{2})}^{2}}, respect y: \dfrac{\partial f}{\partial y}=\displaystyle\frac{-2yx^{3}}{{(x^{2}+y^{2})}^{2}} but then?
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  5. #5
    Bop
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    I have calculate that limit of \dfrac{\partial f}{\partial x} doesn't exist.. How can I continue?
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