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Math Help - Higher order derivatives

  1. #1
    Senior Member Pinkk's Avatar
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    Higher order derivatives

    If u = F(x + g(y)), show that u_{x}u_{xy} = u_{y}u_{xx}.

    I have no idea how to go about this, although I am almost certain it involves the chain rule somehow. Any help would be appreciated.
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  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by Pinkk View Post
    If u = F(x + g(y)), show that u_{x}u_{xy} = u_{y}u_{xx}.

    I have no idea how to go about this, although I am almost certain it involves the chain rule somehow. Any help would be appreciated.
    First note that F is only a function of one variable(lets call it F(t)) and it is composed with a function of two variables t=x+g(y)

    Then by the chain rule we have

    \displaystyle u_x=\frac{\partial F}{\partial t}\frac{\partial t}{\partial x}=F_t(x+g(y))
    \displaystyle u_y=\frac{\partial F}{\partial t}\frac{\partial t}{\partial y}=F_t(x+g(y))\cdot g_y(y)

    This should get you started.
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  3. #3
    Senior Member Pinkk's Avatar
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    Thanks, once I set the argument of F as one variable, the rest followed easily. Thanks again!
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