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A tangent plane to a function \(f(x,y)\) at a point \(P (x_0,y_0)\) is the plane which contains all tangent lines along all traces through \(P\). The following diagram illustrates this idea.
We say that \(f(x,y)\) is differentiable at \(P(x_0,y_0)\) if such a plane exists at \(P\). In this case we can find the plane by using only the trajectories parallel to the \(xz\)-plane and the \(yz\)-plane.
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