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Uncertainty of a derivative

  1. Oct 6, 2015 #1
    y = a / (x - b)3

    a = 77.1 ± 15.2
    b = -1.78 ± 1.18
    x = 21 ± 1

    ---

    y' = -3a / (x - b)4

    How do I find the uncertainty of y'? Thanks.
     
  2. jcsd
  3. Oct 6, 2015 #2

    Mark44

    Staff: Mentor

    You can use the differential of y (dy) to approximate the uncertainty in y. Here, because a, b, and x are varying, what you have is y' = f(a, b, x) = ##\frac {-3a} {(x - b)^4}##.
    So ##\Delta y' \approx dy' = \frac{\partial f}{\partial a} da + \frac{\partial f}{\partial b} db + \frac{\partial f}{\partial x} dx##
    For da, db, dx, use the ± values above.
     
  4. Oct 6, 2015 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You can compute y' for all 8 values a = 77.1 - 15.2, 77.1 + 15.2, b = -1.78 - 1.18, -1.78 + 1.18, x = 21-1, 21+1. Since that might be tedious, you can try to figure out ahead of time where the extreme values of y' will be found. The largest value of ##|y'| = 3 a / |x-b|^4## will occur when the numerator is as large as possible and the denominator is a small as possible. The minimum of ##|y'|## will be found from conditions opposite to those above.

    I am a bit skeptical about using the formula for ##\Delta y'## given in #2, because in this case the ##da, db## are not "small" (but, perhaps, ##dx## is small enough).
     
  5. Oct 6, 2015 #4

    Mark44

    Staff: Mentor

    That thought also occurred to me. Even ##\Delta x## is not all that small, at about 5% of x. The increments for a and b are much worse.
     
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