I am not sure on how to take the partial derivative of
this function with x, y, and z variables.
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This question is designed to fool you. :-)
Yes! If you try to treat this equation as the graph of a function of two variables, you'll wind up with a z on both sides of the equation, which will make you crazy. Instead, just treat the surface as the 2-level surface {(x,y,z): f(x,y,z)=2} where f(x,y,z)=z(1-xe^y cos(z)). Then you take partials with respect to z the same way you take any partial derivative. The equations for the tangent plane of a level surface of a function of three variables is covered n the textbook (remember that?) and in the lecture notes for 10/3/16.
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