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Reciprocal Derivative Identity

  1. Jul 15, 2007 #1


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    Hi. I was just wondering, how can i prove the following identity:

    \frac{{dy}}{{dx}}\frac{{dx}}{{dy}} = 1

    Its nothing that im required to know, but i was just curious, so for all i know, it may be way out of anything that i can mathematically comprehend.

    The best ive been able to do is show that it holds true for some examples that ive tried, but no solid proof.

    Thanks in adavnce,
    Last edited by a moderator: Jul 15, 2007
  2. jcsd
  3. Jul 15, 2007 #2


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    It should be easy using the chain rule. If y= f(x) and f is invertible, then
    x= f-1(y), so that f-1(f(x))= x. Differentiating both sides of that with respect to x,
    [tex]\frac{df^{-1}(y)}{dy}\frac{dy}{dx}= 1[/itex]
    Where I have 'let' y= f(x). Since f-1(y)= x, that is
    [tex]\frac{dx}{dy}\frac{dy}{dx}= 1[/itex]
  4. Jul 15, 2007 #3


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    Ah, easy. Thanks very much for that :smile:
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