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Math Help - Prove the identity by direct differentiation

  1. #1
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    Prove the identity by direct differentiation

    Hi, please have a look at the attached question and the attempt made to solve, but I can't
    The question is Problem 1.2 (b)

    The vectors in my working are underlined, but are bold in the question.

    The thing is, that I don't know where the c springs from in the RHS.
    Here is the question:

    Prove the identity by direct differentiation-football0001.jpg

    My attempt at solving:
    Prove the identity by direct differentiation-football0002.jpg

    Thanks for the help
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  2. #2
    MHF Contributor
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    Re: Prove the identity by direct differentiation

    Just in case a picture helps...



    ... 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 t), and the straight dashed line (and in this more nested case also the squiggly line) similarly but with respect to the dashed (or squiggly) balloon expression (the inner function of the composite which is subject to the chain rule).



    ... is the product rule, where, again, straight continuous lines are differentiating downwards with respect to t.



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  3. #3
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    Re: Prove the identity by direct differentiation

    Thank you very much. I think that this teaches me that there doesn't need to be a very "straight-forward" or "systematic" solution to a problem - that we have to invent stuff according to our needs..
    Am I correct?

    Thank you again for the detailed (and very neat) working.
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  4. #4
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    Re: Prove the identity by direct differentiation

    Er, thank you! I'll hope that 'neat' means not lacking in 'system' and 'straight-forwardness'. Others may offer a more conventional working, anyhow.

    Cheers

    Tom
    Last edited by tom@ballooncalculus; October 12th 2013 at 09:57 AM.
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