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Thread: Partial differentiation question

  1. #1
    s3a is offline
    Super Member
    Nov 2008

    Partial differentiation question

    I am trying to do the first part of #12. The answer is -19.

    I'm trying things like:
    ∂f/∂s = ∂F/∂u = ∂F/∂x * ∂x/∂u + ∂F/∂y * ∂y/∂u
    but then I am stuck.

    Could someone please show me how to do this properly?

    Any help would be greatly appreciated!
    Thanks in advance!
    Attached Thumbnails Attached Thumbnails Partial differentiation question-q12.jpg  
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  2. #2
    MHF Contributor
    Oct 2008

    Re: Partial differentiation question

    Just in case a picture helps...

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


    is the chain rule for two inner functions, i.e...

    $\displaystyle \frac{d}{dx}\ f(u(x), v(x)) = \frac{\partial f}{\partial u} \frac{du}{dx} + \frac{\partial f}{\partial v} \frac{dv}{dx}$

    Or rather, in this example...

    $\displaystyle \frac{d}{ds}\ f(x(s), y(s)) = \frac{\partial f}{\partial x} \frac{dx}{ds} + \frac{\partial f}{\partial y} \frac{dy}{ds}$

    As with...

    ... the ordinary chain rule, straight continuous lines differentiate downwards (integrate up) with respect to s, and the straight dashed line similarly but with respect to the (corresponding) dashed balloon expression which is (one of) the inner function(s) of the composite expression.

    We need a similar (double) diagram to differentiate with respect to t.

    You can simplify the picture by losing many of the labels. Indeed you might have just values in most of the balloons.

    Then do similar for dt, where the continuous lines mean with respect to t, and s is held constant.

    Then do similar for the second derivatives.


    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Last edited by tom@ballooncalculus; Nov 8th 2011 at 04:04 AM. Reason: typo
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