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Math Help - Implicit Function

  1. #1
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    Implicit Function

    I have the following implicit equation as functions of x and y:

    xu+yv+(u^2)-(v^2)-1=0
    2xyuv-1=0

    I found using Jacobian matrices etc. that the partial derivative of u with respect to x is:
    -u(xu-yv+(2v^2))/(x(xu+(2u^2)-yv+(2v^2))

    How do you find the partial derivative of this (ie the second partial derivative d^2u/dx^2, the second partial derivative of u with respect to x)?


    Thank you for any help or clarification.
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  2. #2
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    OK. Just answer this.

    du/dx = partial derivative of u with respect to x =2u

    Does d^(2)u/dx^2 = second partial derivative of u with respect to x = 2?

    Is this right?

    Can I just take the partial derivative of the already calculated partial derivative to find the second derivative?
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  3. #3
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    Quote Originally Posted by SwedishMan View Post
    OK. Just answer this.

    du/dx = partial derivative of u with respect to x =2u

    Does d^(2)u/dx^2 = second partial derivative of u with respect to x = 2?

    Is this right?
    No.
    If u=f(x,y)
    Them,
    (2u)_x=2u_x
    Not,
    (2u)_x=2.
    However, it CAN happen if u=x+f(y)
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  4. #4
    Super Member Rebesques's Avatar
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    You cannot calculate the derivatives explicitly - or else the Implicitly function theorem would be pointless - only at specific points.

    In this case, we get

    u_x+2uu_x-2v+u=0, \ y(uv+xu_xv+xuv_x)=0

    and use the conditions given -something like u(x_0,y_0)=(a,b), v(x_0,y_0)=(c,d) - to solve algebraically this system for u_x, v_x. Same for the other derivatives.
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