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Dynamic operation modes of scanning force microscopy based on probe resonance frequency detection are very successful methods to study force-related properties of surfaces with high spatial resolution. There are well-recognized approaches to measure vertical force components as well as setups sensitive to lateral force components. Here, we report on a concept of bidirectional force gradient microscopy that enables a direct, fast, and quantitative real space mapping of force component derivatives in both the perpendicular and a lateral direction. It relies solely on multiple-mode flexural cantilever oscillations related to vertical probe excitation and vertical deflection sensing. Exploring this concept we present a cantilever-based sensor setup and corresponding quantitative measurements employing magnetostatic interactions with emphasis on the calculation of mode-dependent spring constants that are the foundation of quantitative force gradient studies.