We describe a method we call core extraction for measuring the speed distributions of products from photoinitiated bimolecular reactions for the purpose of determining state-to-state differential cross sections. Core extraction is demonstrated; by determination of the state-to-state differential cross section for the reaction Cl+CH4(upsilon(3)=1)+-->HCl(upsilon=1, J=1)+CH3. The method of core extraction measures three-dimensional projections of the velocity distribution using a time-of-flight mass spectrometer equipped with a mask to reject off-axis scattered products. This three-dimensional projection is then converted to a state-to-state differential cross section via simple transformations. Competition between instrumental resolution and signal in core extraction is discussed, and the behavior of our system is checked with simple velocity distributions that result from photodissociation of Cl-2. Core extraction is compared with other methods for the measurement of state-resolved differential cross sections.