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 X0 = (x0, y0, z0) the expanded form is:
fx(X0)(x-x0) + fy(X0)(y-y0) + fz(X0)(z-z0)
+ fxy(X0)(x-x0)(y-y0) + fxz(X0)(x-x0)(z-z0)
+ fxx(X0)(x-x0)2/2 + fyy(X0)(y-y0)2/2 + fzz(X0)(z-z0)2/2
points out you may want to look at Taylor's theorem
for a more general form of approximation.