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differentiation question

Discussion in 'Mathematics' started by fudgesweets, Jun 18, 2011.

  1. fudgesweets

    fudgesweets New commenter

    I have been trying a step paper and was wondering,

    d / du [dy/dx]= d / dx [dy/du]

    is it correct to say the above is true?
     
  2. fudgesweets

    fudgesweets New commenter

    I have been trying a step paper and was wondering,

    d / du [dy/dx]= d / dx [dy/du]

    is it correct to say the above is true?
     
  3. Seems reasonable but I am not sober so who knows

    Would one say d2y by dxdu
     
  4. From my experience, I'd say it is correct to say that, since it boils down to differentiating y with respect to 2 independent variables x and u. Thus, it shouldn't make any difference with whichever of the two you decide to start with.
     
  5. With suitably generous conditions put on the functions being differentiated, this is known as the mixed derivative theorem. You have to work rather hard to find a bizarre function as a counterexample if the conditions are relaxed.
     
  6. In the context of a STEP paper, I'd be nervous about assuming u and x are independent. (Not because STEP would be particularly tricky about u and x, but because I'd think the most likely way this came about was from some kind of substitution u = f(x)).

    I don't think I've ever seen a STEP question that was supposed to be solved by differentiating w.r.t. two different variables.
     
  7. Of course D franklin, I totally agree with you!! There should be no reason to assume in mathematics especially at this level, I would imagine the complete question would have the elements for the student to decide if the variables are independent or not. I have actually seen a differentiation question in STEP of the form y=ue^x (subquestion) where u is a function of x. In this particular case, it doesn't really matter which variable you start differentiating with. In any case, there is no basis to generalise the assertion that d/du(dy/dx) is always the same as d/du(dy/dx)
     
  8. Sure, you get lots of questions with u dependent on x (or vice versa). But I don't think I've ever seen a genuine "two indep variables" differentiation question (one where you could justifiably use \frac).Without that, I don't think the identity the OP wishes to assume is valid.
    (Of the several hundred questions I've done, I think I recall one question where you might reasonably want to diff under the integral sign, but even then it was far from necessary).
     

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