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Implicit Differentiation

  1. Oct 5, 2015 #1

    K41

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    I have an equation:

    r^2 = x^2

    So I found out dr/dx = x/r.

    But when I try to find the second derivative, I get d2r/dx2 = -x^2/r^3 when the text says it should be (r^2 - x^2)/r^3.

    Can anyone help? My working out:

    r^2 - x^2 = 0
    r^2 = x^2.
    Assume r is a function of x.
    rr' = x (first derivative found correctly)
    rr'' + r'(x/r) = 1 (apply chain rule and sub in answer for first derivative)
    rr'' + x^2/r^2 = 1 (sub in first derivative)

    So where have I gone wrong?
     
  2. jcsd
  3. Oct 5, 2015 #2

    Mentallic

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    And then where do you go from that last line?
     
  4. Oct 5, 2015 #3

    BvU

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    Only in the very last steps (after your last line):
    rr'' + x2/r2 = 1 ⇔
    rr'' = 1 - x2/r2
    r'' = 1/r - x2/r3
    r'' = ( r2 - x2 ) / r3
     
  5. Oct 5, 2015 #4

    K41

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    Haha, you won't believe what I was doing. Instead of subtracting both sides, I was doing a division (for reasons not clear to me or anyone of the known realm)...

    GGGAAAAAHHHH

    Thanks!
     
  6. Oct 5, 2015 #5

    BvU

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    I believe you. You are not the only one....
     
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