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Inverting a derivative

  1. Jan 12, 2010 #1
    Suppose that we have (on some domain) a 1 - 1 function y(x). So we can alternatively write x(y). Consider a point x_0 and let y_0 = y(x_0). Suppose

    [tex]\frac{dy}{dx}(x_0) = f(x_0)[/tex]

    Is it always true that

    [tex]\frac{dx}{dy}(y_0) = \frac{1}{f(x(y_0))}[/tex]

    ? If not, under what conditions might it be false?
     
  2. jcsd
  3. Jan 12, 2010 #2

    HallsofIvy

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    Science Advisor

    As long as f is one-to-one and so has an inverse function, that is true. As usual, you can prove properties where you are treating the derivatve as if it were a fraction (here that dx/dy= 1/(dy/dx)) by going back before the limit of the difference quotient, using the fact that the difference quotient is a fraction and then taking the limit again.
     
  4. Jan 12, 2010 #3
    Awesome. Thanks!
     
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