I am taking computer vision course, where I learned about Sobel operator for detecting edges. I somewhat understand how it works as described here. But I have a basic confusion. The operator can be divided along $x$ and $y$ axis (as $G_x$ and $G_y$). But these divided operators are finding derivative along one direction, then why are they 2-dimensional?
Actually the answer is already provided in the wiki page as follows:
"Since the Sobel kernels can be decomposed as the products of an averaging and a differentiation kernel, they compute the gradient with smoothing..."
Indeed those 1-D differentiation and averaging kernels are pre-multiplied to form the 2D final kernels to compute directional derivaties.
Hence, even though as you correctly expected the row and column derivates can be computed via 1D kernels, the averaging function being utilized into, renders them to be 2D.