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Math Help - Jacobian Determinants (Implicit differentiation)

  1. #1
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    Question Jacobian Determinants (Implicit differentiation)

    The equations,

    x^{2} + y +3s^{2} + s = 2t-1, y^{2} - x^{4} +2st +7 = 6s^{2}t^{2}

    define s and t as functions of x and y. Find \frac{\partial s}{\partial x} when s = 0 and t = 1. Assume x > 0.

    I've gone through the whole process defining the functions F and G and using Jacobian determinants to solve for \frac{\partial s}{\partial x} but I'm still stuck with some x's in my answer that I can't figure out eliminate.

    The answer listed is -16, but I after all my computation I came up with 2x^3.

    You can see that if you define the first equation as a function F(s,t,x,y) = 0, and G(s,t,x,y) = 0 the partial with respect to x of equation G (-4x^3) won't go away, even after evaluating s and t at 0 and 1 repsectively.

    Any ideas/suggestions?
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  2. #2
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    Your answer \frac{\partial s}{\partial x} = 2x^3 (at the point where (s,t) = (0,1)) is correct. To find the value of x, just put s=0 and t=1 in the original equations. You get

    x^2+y = 1,\quad y^2-x^4 +7 = 0.

    After factorising x^4-y^2 as (x^2+y)(x^2-y) and using the given information that x>0, you should be able to solve those equations to get x=2. That gives the answer as \frac{\partial s}{\partial x} = 16, not –16 as the book claims. It looks as though the book is wrong.
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