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I Chain rule and change of variables again

  1. Jan 11, 2017 #1
    We start with:


    And we want to consider x as function of y instead of y as function of x.

    I understand this equality:

    dy/dx = 1/ (dx/dy)

    But for the second order this equality is provided:

    d2y/dx2 =- d2x/dy2 / (dx/dy)3

    Does anybody understand where is it coming from? The cubic term in the denominator looks quite strange, I don't know how to understand that equation.

    ANy help is welcome.
  2. jcsd
  3. Jan 11, 2017 #2


    Staff: Mentor

    Write your second equation above as ##dy/dx = (dx/dy)^{-1}##
    Now take the derivative with respect to x of both sides:
    ##d^2y/dx^2 = (-1) (dx/dy)^{-2} \cdot d/dx(dx/dy)##
    On the right side above, I'm using the chain rule.
    Is that enough of a start?
  4. Jan 11, 2017 #3
    It looks like a great start but I have a question.

    If we have x as function of y, dx/dy will be a function of y too, WHat is the meaning of deriving this respect to x?
    X is the independent variable now, Does it make sense to differenciate a function respect itself?

    Maybe I am confused and I should go to bed.
  5. Jan 11, 2017 #4


    Staff: Mentor

    You can differentiate it with respect to x (not derive it).
    No, if x is a function of y, x is the dependent variable, and y is the independent variable.
    In the last part of what I wrote I have ##d/dx(dx/dy)##. That's the same as ##\frac d {dy} \left(\frac {dx}{dy} \right)\cdot \frac{dy}{dx}##, with the chain rule being applied once more.
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