The document proves that a continuous function f from R2 to R that has a tangent plane at a point (x1,y1,f(x1,y1)) must also have a directional derivative at that point in all directions. It does this by showing that the tangent plane equation involves the partial derivatives fx and fy, and the directional derivative is defined using those same partial derivatives. Therefore, if a tangent plane exists, the partial derivatives must also exist, and the function has a directional derivative.