Optical differential operation based on the photonic spin Hall effect(SHE) has attracted extensive attention in image processing of edge detection, which has advantages of high speed, parallelism, and low power consumption. Here, we theoretically demonstrate tunable optical differential operation in a four-layered nanostructure of prism-graphene-air gap-substrate. It is shown that the spatial differentiation arises inherently from the photonic SHE. Furthermore, we find that the transverse spin-Hall shift induced by the photonic SHE changes dramatically near the Brewster angle with the incident angle increases at a telecommunication wavelength. Meanwhile, the Fermi energy of graphene and the thickness of the air gap can affect the transverse spin shift. Interestingly, we can easily adjust the Fermi energy of graphene in real time through external electrostatic field biasing, enabling fast edge imaging switching at a telecommunication wavelength. This may provide a potential way for future tunable spin-photonic devices, and open up more possible applications for artificial intelligence, such as target recognition, biomedical imaging, and edge detection.