A quadratic approximation in three variables approximates a 3 dimensional function. This approximation involves partial derivatives.

If you are approximating the f, the function taking in a vector at location **X**_{0} = (x_{0}, y_{0}, z_{0}) the expanded form is:

Q(**X**) =
f(**X**_{0}) +
f_{x}(**X**_{0})(x-x_{0}) + f_{y}(**X**_{0})(y-y_{0}) + f_{z}(**X**_{0})(z-z_{0})
+ f_{xy}(**X**_{0})(x-x_{0})(y-y_{0}) + f_{xz}(**X**_{0})(x-x_{0})(z-z_{0})
+ f_{yz}(**X**_{0})(y-y_{0})(z-z_{0})
+ f_{xx}(**X**_{0})(x-x_{0})^{2}/2 + f_{yy}(**X**_{0})(y-y_{0})^{2}/2 + f_{zz}(**X**_{0})(z-z_{0})^{2}/2

Swap points out you may want to look at

Taylor's theorem for a more general form of approximation.