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Simple Differentiation - Is this legal / correct method?

  1. Oct 1, 2015 #1
    I have the equation:
    x = 2*L*sin(θ/2) and in my lecture notes they have converted it to: ϑx = L*cos(θ/2)*ϑθ

    Is it correct to do the following to get this answer?

    x = 2*L*sin(θ/2)
    x = 2*L*sin(θ/2)*(ϑ(θ/2)/ϑx)
    x*ϑx = 2*L*sin(θ/2)*ϑ(θ/2)
    1*ϑx = (1/2)*2*L*cos(θ/2)*ϑθ
    ϑx = L*cos(θ/2)*ϑθ

    My problem is I don't see how you can keep ϑx and ϑθ after the differentiation operation has been done, and if it is correct to be able to separate ϑ(θ/2) into (1/2)*ϑθ ?

    Any help would be appreciated, Thanks.
     
  2. jcsd
  3. Oct 1, 2015 #2

    SteamKing

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    Staff Emeritus
    Science Advisor
    Homework Helper

    It's called implicit differentiation:

    http://www.sosmath.com/calculus/diff/der05/der05.html

    This approach is a little more long-winded, but it's good you get the same result.
    You are treating dx and dθ as differential quantities here. Doing d(θ/2) = (1/2)dθ is perfectly acceptable.

    Haven't you studied integration by parts or the use of u-substitution to solve integrals yet?
     
  4. Oct 2, 2015 #3

    Mark44

    Staff: Mentor

    I'm not sure what character you used, but the above should be dx = L * cos(θ/2) dθ
    No. Differentiate both sides with respect to x.
    From this you get 1 = 2 * L cos(θ/2) * d/dx(θ/2), or
    1 = 2 * L cos(θ/2) * 1/2 * dθ/dx

    You can then solve algebraically for dθ/dx.
    You can work with differentials: d(θ/2) = (1/2) dθ
    The chain rule is in play here. ##d(θ/2) = \frac{d(θ/2)}{dθ} \cdot dθ = (1/2) dθ##
     
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