Operations like img$*$matrix are, in a generic, all convolutions. And derivatives are instance of convolutions. But convolutions are much more generic. When looking at edges, one can use derivatives at different orders. So you can have different such matrices representing different derivatives. If you convolve two such matrices representing a first derivative, you can get matrix$*$matrix, which is just another kind of (convolution) matrix, but now representing a second derivative (under some conditions I won't detail now).
[EDIT after Atul Ingle] The choice of the coefficients in img drives its behavior. On a discrete image, img can emulate different discretized derivative behaviors, potentially in different directions. Discretizing oriented operators is a complex operations. For instance, you can find here at least three different $3\times 3$ versions for the discrete Laplacian:
Here, are two recent references if you want to dig deeper:
This paper proposes a corner detector and classifier using anisotropic
directional derivative (ANDD) representations. The ANDD representation
at a pixel is a function of the oriented angle and characterizes the
local directional grayscale variation around the pixel. The proposed
corner detector fuses the ideas of the contour- and intensity-based
detection. It consists of three cascaded blocks. First, the edge map
of an image is obtained by the Canny detector and from which contours
are extracted and patched. Next, the ANDD representation at each pixel
on contours is calculated and normalized by its maximal magnitude. The
area surrounded by the normalized ANDD representation forms a new
corner measure. Finally, the nonmaximum suppression and thresholding
are operated on each contour to find corners in terms of the corner
measure. Moreover, a corner classifier based on the peak number of the
ANDD representation is given. Experiments are made to evaluate the
proposed detector and classifier. The proposed detector is competitive
with the two recent state-of-the-art corner detectors, the He & Yung
detector and CPDA detector, in detection capability and attains higher
repeatability under affine transforms. The proposed classifier can
discriminate effectively simple corners, Y-type corners, and higher
order corners.
We describe the design of finite-size linear-phase separable kernels
for differentiation of discrete multidimensional signals. The problem
is formulated as an optimization of the rotation-invariance of the
gradient operator, which results in a simultaneous constraint on a set
of one-dimensional low-pass prefilter and differentiator filters up to
the desired order. We also develop extensions of this formulation to
both higher dimensions and higher order directional derivatives. We
develop a numerical procedure for optimizing the constraint, and
demonstrate its use in constructing a set of example filters. The
resulting filters are significantly more accurate than those commonly
used in the image and multidimensional signal processing literature.
