I don't have didactic examples at hand, but e.g. [1] or [2]. IIRC [1] uses the Laplace operator (second-order spatial derivate) and [2] uses a linear solve inside the forward model through which differentiation is certainly possible but pretty cumbersome in practice.

[1] https://www.nature.com/articles/s41598-019-52283-6

[2] https://doi.org/10.1117/1.JMI.4.3.034005