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Evaluate dy/dt for each of the following

  1. Jun 28, 2008 #1
    1. The problem statement, all variables and given/known data
    Assume x and y are functions of t. Evaluate dy/dt for each of the following.

    xy-5x+2y^3=-70, dx/dt=-5 x=2 y=-3

    2. Relevant equations
    n/a


    3. The attempt at a solution
    I used x' and y' for dx/dt and dy/dt

    I found the derivative of the equation, using the product rule for xy

    (x)(yy')+(y)(xx')-5+6y^2=0

    From here, my problem was figuring out how to simplify the equation...
    (x)(yy')=5-6y^2-yxx'
    y'=

    The answer was -5/7...I'm pretty sure I messed up on solving the equation for 0 and y'
     
  2. jcsd
  3. Jun 28, 2008 #2

    dynamicsolo

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    This is not differentiated correctly with respect to the third variable. When dealing with the product rule for the xy term, think of x and y as if both were functions of t. This would have to be

    d/dt (xy) = (dx/dt)·y + x·(dy/dt),

    or x'y + xy' , as you have been writing it. By the same token, the next term would have the derivative -5x' .

    Keep in mind that you are not differentiating with respect to x, as you were in your other implicit differentiation problems, but with respect to a variable whichi is not directly (we say "explicitly") represented in the original equation.
     
  4. Jun 29, 2008 #3
    So I messed up my product rule of xy then?
     
  5. Jun 29, 2008 #4

    Defennder

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    Yeah I believe so. What is the derivative of xy with respect to t?
     
  6. Jun 29, 2008 #5
    wouldn't it be: x(dy/dt)+y(dx/dt) for the derivative of xy
     
  7. Jun 29, 2008 #6

    dynamicsolo

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    That's correct (and is mentioned in post #2). You need to differentiate the remaining terms on the left-hand side of the original equation in the same fashion. So d/dt (5x) and d/dt (2y^3) are not simply 5 and 6y^2, but...?
     
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