Apparently, the Schwartz kernel theorem states that all linear operators can be represented as integral transforms (but only if you use generalized functions such as the dirac delta as kernels.) Representing the derivative operator as an integral transform would be really useful because then when defining strange and new generalizations of integrals one could automatically derive an appropriate generalization of derivatives to fit.
Yes, you can - but (somewhat disappointingly) the integral kernel for the derivative is just
$$ k(x,y) = -\delta^\prime(x-y) $$ where $\delta^\prime$ is the derivative of the delta distribution, defined in the distributional sense i.e.
$$ \langle \delta^\prime(x-y),\varphi(y)\rangle = -\langle \delta(x-y),\varphi^\prime(y)\rangle = -\varphi^\prime(x) $$
This is occasionally useful for manipulating expressions symbolically. Also look into the single and double layer potential which express a similar idea for directional derivatives across surfaces and is a very useful concept in PDE.