# derivative matrix and the Dirac delta distribution

For a project I'm working on, I was working with the following equation

$$w(x) = \int k(x,y)v(y)dy$$

I noticed that if I choose

$$k(x,y) = -\delta'(x-y)$$

Then we probably get (I haven't touched Dirac delta functions for years, please correct me if I am wrong.).

$$v'(x) = -\int \delta'(x-y) v(y)dy \tag{1}$$

Where $$\delta'$$ is the derivative of the Dirac delta distribution as detailed here and $$v'(x)$$ is the derivative of $$v(x)$$.

My question is: Would it be possible to discretize (1) using some technique to get different finite difference schemes for derivatives? Presumably different definitions of $$\delta$$ using different limit of sequences of functions would lead to different finite difference schemes.

Something like

$$v' = Pv$$

Where $$P$$ is a matrix, $$v$$ is a vector whose entries are values of $$v(x)$$ at equispaced locations and and $$v'$$ is a vector containing the finite difference approximation to the derivative.