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Homework Help: Verify partial derivatives

  1. Sep 28, 2010 #1
    1. The problem statement, all variables and given/known data
    f(x,y)=2Sin x Cos y
    g(x,y) = 2Cos x Sin y
    verify that d(fg)/dx = g(x,y) df/dx + f(x,y) dg/dx


    3. The attempt at a solution
    first of all I worked out the partials derivatives in respective to x and y, for both functions
    df/dx = 2Cos x (but I've a gut feeling that it should be 2Sin x * Cos x)
    df/dy = -Sin y (but I've a gut feeling that it should be Cos y * -Sin y)
    dg/dx=-2Sin x (but I've a gut feeling that it should be 2Cos x * -Sin x)
    dg/dy=Cos y (but I've a gut feeling that it should be Sin y * Cos y)

    but I don't get how to verify d(fg)/dx = g(x,y) df/dx + f(x,y) dg/dx
    can someone walk me through what does the question is asking about?

    Thanks
     
  2. jcsd
  3. Sep 28, 2010 #2
    Just do what the problem asks you.

    a) Multiply f and g toghether as they are, then derive for x.

    b) Then compute g multiplied by df/dx plus f multiplied by dg/dx

    Check if a) and b) are the same.
     
  4. Sep 28, 2010 #3

    HallsofIvy

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    Neither of those is correct. The derivative of 2A Sin x, where A is a constant, is 2A Cos x. Since in taking a partial derivative, you treat the other variable as a constant, the partial derivative of 2 Sin x Cos y is 2 Cos x Cos y.

    Same applies to each of those. (But you don't need the derivatives with respect to y for this problem.)

    but I don't get how to verify d(fg)/dx = g(x,y) df/dx + f(x,y) dg/dx
    can someone walk me through what does the question is asking about?

    Thanks[/QUOTE]
    fg= 4 Sin x Cos x Sin y Cos y. The left side of your equation is the derivative of that with respect to x.
     
  5. Sep 28, 2010 #4

    phyzguy

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    You're not doing the partial derivatives correctly. When I take the partial derivative with respect to x, I treat y like a constant. So you should treat the cos(y) term just like you treat the 2. So the partial of {2 sin(x) cos(y)) with respect to x is 2 cos(x) cos(y), not 2 cos(x).
     
  6. Sep 28, 2010 #5
    fg= 4 Sin x Cos x Sin y Cos y. The left side of your equation is the derivative of that with respect to x.[/QUOTE]

    oh right thank you very much! I think I need more practice on derivatives though.
    another question, say I have the following function
    f(x,y)=x2+y2-xy
    I need to show that u(t)=(cos t; sin t) is a unit vector

    can someone please tell me what steps I need to take to show u(t) is a unit vector of that given function?

    thanks again
     
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