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Math Help - multivariate tangency question

  1. #1
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    multivariate tangency question

    find the tangent plane for the given function:

    \sin(xyz) = x+2y+3z, \ \ (2,-1,0)

    would I have to transfer sin(xyz) to the right then differentiate?
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  2. #2
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    Quote Originally Posted by lllll View Post
    find the tangent plane for the given function:

    \sin(xyz) = x+2y+3z, \ \ (2,-1,0)

    would I have to transfer sin(xyz) to the right then differentiate?
    You can differentiate both sides partially using implicit diferentiation. Eg. Assume z is implicitly defined as a function of y and x and differentiate both sides wrt x:

    \cos (xyz) (yz + xy z_x) = 1 + 3z_x.

    At (2, -1, 0) substitute x = 2, y = -1 and z = 0: \cos (0) (0 - 2z_x) = 1 + 3z_x \Rightarrow -2 z_x = 1 + 3 z_x \Rightarrow z_x = - \frac{1}{5}.

    Do similar to get z_y at (2, -1, 0).

    Remember that a vector normal to the tanget plane is (z_x) i + (z_y) j - k .....
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