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Math Help - [SOLVED] Differentiation

  1. #1
    Member ronaldo_07's Avatar
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    [SOLVED] Differentiation

    u(x,y) = cos(4xy)e^(2x - 2y + 2)

    v(x,y) = sin(4xy)e^(2x - 2y + 2)

    Compute the partial derivatives of u and v with respect to x and y for all values (x, y) and verify that they satisfy the Cauchy-Riemann equations.

    How do I differentiate each?

    for example i got du/dx=4e^(2x - 2y + 2)cos(4xy)
    du/dy=4ye^(2x - 2y + 2)cos(4xy)
    dv/dx=4e^(2x - 2y + 2)sin(4xy)
    dv/dy=4ye^(2x - 2y + 2)sin(4xy)

    Im not sure if this is correct
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  2. #2
    Super Member Aryth's Avatar
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    You're almost right, for each partial derivative you MUST use the product rule, for example:

    \frac{\partial{u}}{\partial{x}} = \frac{\partial{[\cos{(4xy)}]}}{\partial{x}}e^{(2x^2 - 2y^2 - 2)} + \cos{(4xy)}\frac{\partial{[e^{(2x^2 - 2y^2 - 2)}}]}{\partial{x}}

     = -4\sin{(4xy)}e^{(2x^2 - 2y^2 - 2)} + 4x\cos{(4xy)}e^{(2x^2 - 2y^2 - 2)}

    = 4e^{(2x^2 - 2y^2 - 2)}[x\cos{(4xy)} - \sin{(4xy)}]

    You have to differentiate them as such to arrive at answers that you can use for the CR equations.
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