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Math Help - Another Directional Derivative

  1. #1
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    Another Directional Derivative

    I'm new at working with directional derivatives. I'm stuck on this problem. Would someone mind helping me with the "Point" and "Direction" parts of the problem. I'm just looking for a place to start, not for someone to do the problem for me. Thanks.

    In each of the following, find the directional derivative of f at P in the given direction.

    1. f(x,y)=xy(1-x^2-y^2), P=any point on the unit circle, Direction: in the direction of OP.

    I found that the gradient of f is: <y(1-x^2-y^2)-2yx^2, x(1-x^2-y^2)-2xy^2>, so I'm stuck on finding P and the direction.
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  2. #2
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    Re: Another Directional Derivative

    Quote Originally Posted by dbakeg00 View Post
    In each of the following, find the directional derivative of f at P in the given direction.
    1. f(x,y)=xy(1-x^2-y^2), P=any point on the unit circle, Direction: in the direction of OP.
    I found that the gradient of f is: <y(1-x^2-y^2)-2yx^2, x(1-x^2-y^2)-2xy^2>, so I'm stuck on finding P and the direction.
    If P: (a,b) then \overrightarrow {OP}=<a,b> and you know that a^2+b^2=1.
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  3. #3
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    Re: Another Directional Derivative

    Ok, I did the problem, would you mind checking for me? Thanks

    So I changed [the format] of f (x,y) = (xy-x^3y-xy^3)
    and P=(a,b) & OP=<a,b>

    Now I have the gradient of f=<y-3x^2y-y^3, x-x^3-3xy^2>

    Duf(x,y)=(a)(y-3x^2y-y^3)+(b)(x-x^3-3xy^2)

    Duf(a,b)=(a)(b-3a^2b-b^3)+(b)(a-a^3-3ab^2)

    Duf(a,b)= (ab-3a^3b-ab^3) + (ab-a^3b-3ab^3)

    Duf(a,b)= 2ab-4a^3b-4ab^3

    Duf(a,b)= ab-2a^3b-2ab^3

    Duf(a,b)= ab(1-2a^2-2b^2)
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