A new edge detection technique based on derivative algorithm in arbitrary directions is developed. Discrete representation of derivative for grey image and determination of optimal response direction are important to edge detection techniques. The mathematic model of derivative for grey image was represented, and the first derivative and the second derivative algorithms in arbitrary directions were derived from the basic derivative definition. Based on the characteristic analysis of image edge, the grey-level gradients and Laplace transforms in multi-directions or full directions of view were calculated, and the maximums determined the output values for the edge magnitude image. The new algorithm is demonstrated to be valid by detecting the edge of normal image and some advantages of the algorithm are represented over the conventional edge operators.