The problem of scale is of fundamental interest in image processing, as the features that we visually perceive and find meaningful vary significantly depending on their size and extent. It is well known that the strength of a feature in an image may depend on the scale at which the appropriate detection operator is applied. It is also the case that many features in images exist significantly over a limited range of scales, and, of particular interest here, that the most salient scale may vary spatially over the feature. Hence, when designing feature detection operators, it is necessary to consider the requirements for both the systematic development and adaptive application of such operators over scale- and image-domains.We present a new approach to the design of scalable derivative edge detectors, based on the finite element method, that addresses the issues of method and scale adaptability. The finite element approach allows us to formulate scalable image derivative operators that can be implemented using a combination of piecewise-polynomial and Gaussian basis functions. The issue of scale is addressed by partitioning the image in order to identify local key scales at which significant edge points may exist. This is achieved by consideration of empirically designed functions of local image variance.