Find the tangent plane to the surface x^2+y^2+xyz=4+z^3(y-2x) at the point (1, 1, 1).
So the equation of the tangent plane is z-z0=f'x(x-x0)+f'y(y-y0).
I find the partials using implicit differentiation:
f'x=(2x)/(3yz^2-6z^2-y) @ (1,1,1) = -1/2
f'y=(2y)/(3z^2-6xz^2-x) @ (1,1,1) = -1/2
Then the equation of the plane (ie plugging into the above equation) is z+(1/2)x+(1/2)y=2
But the answer is 5x+2y+4z=11
I have two question from here:
1, Where am I going wrong in the implicit differentiation? Wolfram gives me a different result.
2, I know using this method is incorrect. Why is that? Why must I use the gradient vector to the level surface instead of implicitly deriving?

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This is calc III by the way

I'm not so confused about the problem itself but rather why these two methods are not equivalent

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