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Math Help - Calculating 2nd Derivative of an implicit function using chain rule.

  1. #1
    Senior Member bugatti79's Avatar
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    Calculating 2nd Derivative of an implicit function using chain rule.

    Dear all,

    I get stuck when I try to determine U_x_x where u(x(\xi,\eta))

    I calculate u_x=\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial u}{\partial \eta}\frac{\partial \eta}{\partial x}

    u_x_x=\frac{\partial^{2} u}{\partial \xi^{2}}\frac{\partial^{2} \xi}{\partial x^{2}}...... Something is telling me this is wrong but I dont know how to proceed any further

    Any suggestions will be appreciated.

    Thanks
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  2. #2
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by bugatti79 View Post
    Dear all,

    I get stuck when I try to determine U_x_x where u(x(\xi,\eta))

    I calculate u_x=\frac{\partial u}{\partial \xi}\frac{\partial \xi}{\partial x}+\frac{\partial u}{\partial \eta}\frac{\partial \eta}{\partial x})

    ok, I calculate

    u_x_x=\frac{\partial^{2}\xi}{\partial x^{2}}\frac{\partial u}{\partial \xi}}+\frac{\partial \xi}{\partial x}(\frac{\partial^{2} u}{\partial \xi^{2}}\frac{\partial \xi}{\partial x}+\frac{\partial^{2} u}{\partial \eta \partial \xi}\frac{\partial \eta}{\partial x})

    I think u_x_x involves a chain rule within a product rule. Can anyone confirm this? Thanks
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  3. #3
    Senior Member bugatti79's Avatar
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    Quote Originally Posted by bugatti79 View Post
    ok, I calculate

    u_x_x=\frac{\partial^{2}\xi}{\partial x^{2}}\frac{\partial u}{\partial \xi}}+\frac{\partial \xi}{\partial x}(\frac{\partial^{2} u}{\partial \xi^{2}}\frac{\partial \xi}{\partial x}+\frac{\partial^{2} u}{\partial \eta \partial \xi}\frac{\partial \eta}{\partial x})

    I think u_x_x involves a chain rule within a product rule. Can anyone confirm this? Thanks
    Sorry I forgot to put in the second chunk, the complete equation is

    u_x_x=\frac{\partial^{2}\xi}{\partial x^{2}}\frac{\partial u}{\partial \xi}}+\frac{\partial \xi}{\partial x}(\frac{\partial^{2} u}{\partial \xi^{2}}\frac{\partial \xi}{\partial x}+\frac{\partial^{2} u}{\partial \eta \partial \xi}\frac{\partial \eta}{\partial x}) +\frac{\partial^{2}\eta}{\partial x^{2}}\frac{\partial u}{\partial \eta}}+\frac{\partial \eta}{\partial x}(\frac{\partial^{2} u}{\partial \eta^{2}}\frac{\partial \eta}{\partial x}+\frac{\partial^{2} u}{\partial \xi \partial \eta}\frac{\partial \xi}{\partial x})

    Thanks
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