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Math Help - Partial Derivative

  1. #1
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    Partial Derivative

    Can someone show me how to get the final answer of cos2t for


    Find the derivative of w = xy with respect to t along the path x = cost and y = sint.


    I know it's chain rule for partials but i'm still confused on how to obtain that final answer.
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  2. #2
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    Edit nevermind, I got it.

     w = xy

     \frac{dw}{dt} = \frac{\partial w}{\partial x}\frac{dx}{dt} + \frac{\partial w}{\partial y}\frac{dy}{dt}

     \frac{\partial w}{\partial x} = y

     \frac{dx}{dt} = -sin(t)

     \frac{\partial w}{\partial y} = x

     \frac{dy}{dt} = cos(t)

    sub values in
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  3. #3
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    What do you mean? How does that net me the final answer of cos2t?
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  4. #4
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    Well, by subbing in those values, we get:  \frac{dw}{dt} = -ysin(t) + xcos(t)

    you defined earlier that  y = sin(t) and  x = cos(t)

    so by subbing these to values into  \frac{dw}{dt} = -ysin(t) + xcos(t)

    we get:

     \frac{dw}{dt} = -sin^2(t) + cos^2(t)

    using trigonometric identity  sin^2(t) + cos^2(t) = 1

    we can see that  -sin^2(t) + cos^2(t) = -(1 - cos^2(t)) + cos^2(t) = 2cos^2(t) - 1

    and finally we can use trig identity:  2cos^2(t) - 1 = cos(2t)

    Hope the more elaborated explanation makes more sense,
    Last edited by U-God; October 31st 2008 at 03:26 PM.
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