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Math Help - Implicit Differentiation

  1. #1
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    Implicit Differentiation

    x^2+y^2+r^2-2s+13=0

    x^3-y^3-r^3+3s+59=0

    How do I find the partial derivatives of x(r,s) or y(r,s) implicitly? I tried implicit differentiation and I got 2 different answers for either. Can someone show me any of the 4 derivatives step-by-step?
    Last edited by MathGuru; December 11th 2005 at 10:41 AM.
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  2. #2
    Super Member Rebesques's Avatar
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    You can only find the values of the derivatives at specific points. Differentiate both for r, then both for s, and evaluate at (r_0,s_0) to get:

    2x(r_0,s_0)\frac{\partial x}{\partial r}(r_0,s_0)+2y(r_0,s_0)\frac{\partial y}{\partial r}(r_0,s_0)+2r_0=0

    3x^2(r_0,s_0)\frac{\partial x}{\partial r}(r_0,s_0)-3y^2(r_0,s_0)\frac{\partial y}{\partial r}(r_0,s_0)-3r_0^2=0

    2x(r_0,s_0)\frac{\partial x}{\partial s}(r_0,s_0)+2y(r_0,s_0)\frac{\partial y}{\partial s}(r_0,s_0)-2=0

    3x^2(r_0,s_0)\frac{\partial x}{\partial s}(r_0,s_0)-3y^2(r_0,s_0)\frac{\partial y}{\partial s}(r_0,s_0)+3=0


    The values r_0, \ s_0, \ x(r_0,s_0), \ y(r_0,s_0) will be given, so the previous four equations form an algebraic system to be solved for \frac{\partial x}{\partial s}(r_0,s_0), \ \frac{\partial y}{\partial s}(r_0,s_0) .
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