As an image is a function of two (or more) variables it is necessary to define the direction in which the derivative is taken.

For the two-dimensional case we have the horizontal direction, the vertical direction, or an arbitrary direction which can be considered as a combination of the two.

First Order Derivative Filter

Second Order Derivative Filter

Derivative Filter is a type of High pass filter used for sharpening.

First-order derivatives generally produce thicker edges. Second-order derivatives have stronger response to fine detail (e.g., thin lines and points)

First-order derivatives have stronger response to step changes. Second-order derivatives produce double response at step changes

Figure 10