A nonlinear derivative is directly defined in the discrete domain. This derivative is motivated by the asymmetry of pattern in discrete signal as step or edge in 2D signal. Thanks to the special definition (in the discrete domain) of this derivative, pattern can be detected in a univocal way. This derivative is the only one able to perfectly detect and localize ideal edges in image. Beside this fundamental benefit, the derivative has the nice property to reduce noise. Ap- plications to edge detection, noise reduction and noise estimation are described and their performances are studied.