Tangent Plane

Let $(x_0,y_0)$ be any point of a surface function $z = f(x,y)$. Then the surface has a nonvertical tangent plane at $(x_0,y_0)$ with equation

$$z = f(x_0,y_0)+ f_x(x_0,y_0)(x-x_0)+f_y(x_0,y_0)(y-y_0).$$

See also Normal Vector, Tangent, Tangent Line, Tangent Space, Tangent Vector