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Homework Help: Differentiating with multiple variables

  1. Mar 8, 2008 #1
    1. The problem statement, all variables and given/known data
    Find dw/dt. Check result by substitution and differentiation

    w = (x^2 + y^2)^1/2, x = e^2t , y = e^-2t

    2. Relevant equations

    3. The attempt at a solution
    dx/dw = x/(x^2 + y^2)^1/2 dy/dw = y/(x^2 + y^2)^1/2

    Dont really know where to go with it
    Last edited: Mar 8, 2008
  2. jcsd
  3. Mar 8, 2008 #2
    One way would be to substitute for x and y, which would result in a function of t alone.
  4. Mar 8, 2008 #3


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    That's the way they suggest to check the answer. They actually do want you to use partial derivatives, mattb8818. Use the chain rule.
  5. Mar 9, 2008 #4


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    Hi matt! :smile:

    First, it's not dx/dw and dy/dw, it's t'other way up!:

    dw/dx = x/(x^2 + y^2)^1/2 dw/dy = y/(x^2 + y^2)^1/2

    Does that help? :smile:
  6. Mar 9, 2008 #5
    since w is a function of x and y
    and x and y are functions of t
    the chain rule is

    dw/dt = dw/dx dx/dt + dw/dy dy/dt

    compute everything above and you're set
    an easy way to remember the chain rule is to draw a tree diagram

    x y
    t t
  7. Mar 9, 2008 #6
    Thanks for the replies

    I did it with the chain rule and got

    (2xe^2t)/(x^2 + y^2)^1/2 + -(2e^-2t)/(x^2 + y^2)^1/2

    The answer is 2(s)^1/2(sinh4t)/(cosh4t)^(1/2)

    I dont really understand this solution
  8. Mar 9, 2008 #7


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    Hi matt!

    I don't know what you did to get 2(s)^1/2(sinh4t)/(cosh4t)^(1/2). :confused:
    Start from that line (which is correct, except you missed out a "y");
    then re-write it as:
    [(2xe^2t) - (2ye^-2t)]/√(x^2 + y^2)
    (this is both to simplify it, and to lessen the risk of making a mistake)
    and just substitute for x and y …

    so what is the next line? :smile:
    [size=-2](btw, if you type alt-v, it prints √ )[/size]​
  9. Mar 9, 2008 #8
    Ok thanks all

    I am glad I atleast did it right. I looked it up and e^x-e^-x/2 = sinh or something like that, but I guess that was my only mistake so Im happy.
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