In the continuous time-function formalism, and accepting the framework or distributions or generalized functions, the answer is direct. Taking $\delta$ for the Dirac delta function, for a sufficiently well-behaved function $f$:
$$\delta' * f = \delta * f' = f'\,.$$
Therefore, the convolution mask is obvious: it would be the derivative of the Dirac delta. The derivative operator is linear, time-invariant, as for the convolution.
Issues arise in practice when the function is not continuous, not known fully: finding a discrete equivalent to the Dirac delta derivative is not obvious.
Therefore, numerous finite difference approximations have proposed in many domains, to adapt to discrete data, non-uniform sample, knowledge of only one side of the data (causality), disturbances of the measurement. They often combine:
- evaluation of the data on a finite interval support,
- regularization or smoothing,
- optimization so that the result is "close enough" to some expected behavior of the "discrete derivative".
Smoothing and optimization are often performed in a least-square sense with interpolation or extrapolation, and hence yield linear, time-invariant discrete "convolution-like" operators with masks. Solutions are numerous, due to the degrees of freedom of the above (support size, smoothing shape, domain of interpolation). Methods range from Lagrangian, Bessel, Newton-Gregory, Gauss, Sterling interpolating polynomials to FIR filter approximation. Some references are:
Note however that some use non-linear or non-time-invariant or non-space-invariant finite differentiation, for instance in real-time computing to limit instabilities or overshoot (references in
CHOPtrey: contextual online polynomial extrapolation for enhanced multi-core co-simulation of complex systems), or in image processing, like in mathematical morphology. There, finite derivatives vary, or using non-linear min/max operators. They are not implemented by convolutions in that case.