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Math Help - Directional Derivatives

  1. #1
    Super Member Aryth's Avatar
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    Directional Derivatives

    Compute the directional derivative of the following function at the indicated points in the given directions:

    f(x,y) = \log{\sqrt{x^2 + y^2}}, \ (x_0,y_0) = (1,0)

    \bold{v} = \frac{1}{\sqrt{5}}(2\bold{i} + \bold{j})

    I did this:

    \nabla f(\bold{x}) = (\frac{2x}{x^2 + y^2},\frac{2y}{x^2 + y^2})

    \nabla f((1,0)) = (2,0)

    \nabla f((1,0)) \cdot \bold{v} = (2,0) \cdot (\frac{2}{\sqrt{5}},\frac{1}{\sqrt{5}})

    = \frac{4}{\sqrt{5}}

    But I'm not sure if this is right.
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  2. #2
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    But you are correct.
    Way to go.
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  3. #3
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    Opalg's Avatar
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    \log{\sqrt{x^2 + y^2}} = {\textstyle\frac12}\log(x^2+y^2), and when you differentiate it the 1/2 cancels with the 2 from the 2x (or 2y). So your answer is twice as big as it should be. Other than that, it looks correct.
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  4. #4
    Super Member Aryth's Avatar
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    I see, thanks.
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