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Math Help - Directional Derivative Question

  1. #1
    Member CalcGeek31's Avatar
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    Directional Derivative Question

    In what direction does the directional derivative for f(x, y) = x2y at (1, 1) have

    A. Value 1?


    B. Value -2?


    C. Is there a direction for which the value is 4?

    I'll be completely honest, I don't even know where to begin on this question, I have never seen a problem like it before therefore I come to you guys lost and looking for help, anything will be appreciated.
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  2. #2
    MHF Contributor Bruno J.'s Avatar
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    The gradient is \bigtriangledown f (x,y)= (2y, 2x), and at (1,1) we have \bigtriangledown f (1,1)= (2, 2). The directional derivative in the direction of the unit vector u=(a,b) is \bigtriangledown f (1,1) \cdot (a,b)=2a+2b. So if you know that this directional derivative is equal to some value X, you want to solve the system of equations:

    2a+2b=X

    a^2+b^2=1

    for a,b.
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  3. #3
    Member CalcGeek31's Avatar
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    I copy pasted wrong, it is supposed to be f(x,y)=(x^2)y, however, i see the process is the same, so after getting the directional derivative, i set it equal to some value, however, where did the a^2 + b^2 = 1 come from?
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  4. #4
    MHF Contributor Bruno J.'s Avatar
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    Quote Originally Posted by Bruno J. View Post
    ...The directional derivative in the direction of the unit vector u=(a,b) ...
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