In this paper, we give a new demonstration in which it is proved that the symmetric exponential filter is the optimal edge detection filter in the criteria of the signal to noise ratio, localization precision and unique maximum. Then we deduce the first and the second directional derivative operators for symmetric Exponential Filter and realize them by first order recursive algorithm, and propose to detect the edges by maxima of Gradient (GEF), or by the zeros crossing of Second directional Derivative along the gradient direction (SDEF).