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Math Help - Directional derivatives and partial derivatives

  1. #1
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    Directional derivatives and partial derivatives

    Suppose f: R -> R is differentiable and let h(x,y) = f(√(x^2 + y^2)) for x ≠ 0. Letting r = √(x^2 + y^2), show that:

    x(dh/dx) + y(dh/dy) = rf'(r).




    I have begun by showing that rf'(r) = sqrt(x^2 + y^2) * limt->0 (f(r+t) - f(r))/t

    and written out the definition form of the directional derivatives. I cant seem to find a way to equate both sides of the equation. Can anyone help?
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  2. #2
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    Re: Directional derivatives and partial derivatives

    Hey tlawrence.

    What did you calculate dh/dx and dh/dy in terms of the function f^n(r)? (Note: f^n(r) is the nth derivative of the function f with argument r)?
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  3. #3
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    Re: Directional derivatives and partial derivatives

    Also/anyway, f(r) and r(x,y) are differentiable so you can apply the chain rule...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x or y), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).



    Similarly with dh/dy. Then substitute into the left-hand side of the show-sentence.



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