Editing Bicubic interpolation (section)

==Finding derivatives from function values==

If the derivatives are unknown, they are typically approximated from the function values at points neighbouring the corners of the unit square, e.g. using [[finite differences]].

To find either of the single derivatives, <math>f_x</math> or <math>f_y</math>, using that method, find the slope between the two ''surrounding'' points in the appropriate axis. For example, to calculate <math>f_x</math> for one of the points, find <math>f(x,y)</math> for the points to the left and right of the target point and calculate their slope, and similarly for <math>f_y</math>.

To find the cross derivative <math>f_{xy}</math>, take the derivative in both axes, one at a time. For example, one can first use the <math>f_x</math> procedure to find the <math>x</math> derivatives of the points above and below the target point, then use the <math>f_y</math> procedure on those values (rather than, as usual, the values of <math>f</math> for those points) to obtain the value of <math>f_{xy}(x,y)</math> for the target point.  (Or one can do it in the opposite direction, first calculating <math>f_y</math> and then <math>f_x</math> from those.  The two give equivalent results.)

At the edges of the dataset, when one is missing some of the surrounding points, the missing points can be approximated by a number of methods. A simple and common method is to assume that the slope from the existing point to the target point continues without further change, and using this to calculate a hypothetical value for the missing point.