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General chain rule (mulitvariable calculus)

  1. Jun 24, 2013 #1
    1. The problem statement, all variables and given/known data

    question from early transcendentals (Edwards, penny) . chapter 12 partial differentiation
    problem 38

    1/R = 1/R1 + 1/R2 + 1/R3
    R is resistance measured in Ω

    R1 and R2 are 100Ω and increasing with 1Ω/s
    R3 are 200Ω and decreasing with 2Ω/s

    Is R increasing or decreasing at that instant ? at what rate ?


    2. Relevant equations
    3. The attempt at a solution

    so what i have done is this

    R(R,R,R) = (1/R1 + 1/R2 + 1/R3)-1

    R1 = r1 + t
    R2 = r2 + t
    R3 = r3 - 2t

    r is the initial values

    using the chain rule
    dR/dt = dR/dR1 * dR1/dt + dR/dR2 * dR2/dt + dR/dR3 * dR3/dt

    i have probably misunderstood something important here . i get some answers but its not right, can someone explain to me how this works and where im wrong.


    the solution is supposed to be 0.24Ω/s
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Jun 24, 2013 #2

    LCKurtz

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    Hard to tell where you went wrong without seeing more details. Your work would be easier if you take the equation$$
    R^{-1}= R_1^{-1}+R_2^{-1} + R_3^{-1}$$and differentiate that just as it stands implicitly with respect to t. Then put in the values.
     
  4. Jun 24, 2013 #3

    Curious3141

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    Let ##f(R) = \frac{1}{R}##

    Now ##\frac{df(R)}{dt} = f'(R).\frac{dR}{dt}##

    ##f'(R)## is trivial

    Now ##\frac{df(R)}{dt} = \frac{\partial f(R)}{\partial R_1}\frac{dR_1}{dt}+\frac{\partial f(R)}{\partial R_2}\frac{dR_2}{dt}+\frac{\partial f(R)}{\partial R_3}\frac{dR_3}{dt}##

    Can you proceed from there?
     
  5. Jun 24, 2013 #4

    Curious3141

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    BTW, assuming a linear relationship of the three resistances over time is invalid. All you know (or need to know) is that *at that instant*, the rates of change are as given. At another instant, they may be different, and all the resistances may behave in wildly nonlinear fashion.
     
  6. Jun 24, 2013 #5
    but example f(R)/dR1

    how do u do that one. doesn't that require even more substitutions.

    because f(R)=1/R
     
  7. Jun 24, 2013 #6

    Curious3141

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    Since I've defined ##f(R) = \frac{1}{R}## and since ##\frac{1}{R} = \frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}##, you simply have ##f(R)=\frac{1}{R_1}+\frac{1}{R_2}+\frac{1}{R_3}##. You should be able to work out the partial derivatives very easily from that, and if you can't, I suggest a more thorough review of the topic.
     
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