Linear approximation for two variables
To approximate a function of two variables with a tangent plane is the natural extension of approximating a function of one variable with a tangent line. In the same way that zooming in a function of one variable makes it render closer to a straight line, with a tangent plane we see that the level curves become closer to straight parallel lines if we zoom in enough.
(not to scale)
As with single variable functions, a multivariable function has to be continuous and differentiable for us to use the tangent plane approximation. In case it's continuous but not differentiable a plane exists, but it's not the same as the tangent plane because if the function is not differentiable there can't be a tangent plane. For example: [math]\displaystyle{ f(x,y) = \sqrt{x^2 + y^2} }[/math]. A plane at [math]\displaystyle{ (0,0) }[/math] is not going to be horizontal. It's going to be angled in one direction or another.
The tangent plane
In analytical geometry a plane is defined with [math]\displaystyle{ Z = O + t_1\overrightarrow{v_1} + t_2\overrightarrow{v_2} }[/math]. In the vector form each point of it is given by a point of origin, two parameters and two linearly independent vectors. In the general form we have an equation that should have been seen in school at some point [math]\displaystyle{ Ax + By + Cz + d = 0 }[/math].
Assuming the function to be differentiable at a point, we have:
[math]\displaystyle{ f(x_0,y_0) }[/math] the point of origin.
[math]\displaystyle{ (x - x_0) }[/math] and [math]\displaystyle{ (y - y_0) }[/math] two pairs of points, belonging to the function's domain, that give the direction in [math]\displaystyle{ x }[/math] and in [math]\displaystyle{ y }[/math].
[math]\displaystyle{ \frac{\partial f}{\partial x}(x_0,y_0) }[/math] and [math]\displaystyle{ \frac{\partial f}{\partial y}(x_0,y_0) }[/math] the variation in each direction, which corresponds to [math]\displaystyle{ t_1 }[/math] and [math]\displaystyle{ t_2 }[/math] in the vector form.
Therefore, the equation of the tangent plane is:
[math]\displaystyle{ z - f(x_0,y_0) = \frac{\partial f}{\partial x}(x_0,y_0)(x - x_0) + \frac{\partial f}{\partial y}(x_0,y_0)(y - y_0) }[/math]
With this equation we find all points of a tangent plane. We disregard the [math]\displaystyle{ d }[/math] because this plane is not any plane, it's tied to a function of two variables.
