Starting from the radiative transfer equation and its usual boundary conditions, the objective of this work is to design Monte Carlo algorithms estimating the specific intensity spatial and angular derivatives as well as its geometric sensitivity. The present document is structured in three parts, each of them dedicated to a specific derivative of the intensity. Although they are all assembled here in one document each derivative is of interest independently whether it be for radiative transfers analysis or engineering conception. Therefore, they are thought to be written as three different papers and are presented here as such. Estimating derivatives of the specific intensity when solving radiative transfers using a Monte-Carlo algorithm is challenging. Finite differences are often not sufficiently accurate and directly estimating the derivative from a specific Monte-Carlo algorithm can lead to arduous formal or numerical developments. The proposition here is to work from the radiative transfer equation and its boundary conditions to design a physical model for each derivatives. Only then Monte-Carlo algorithms are built from the derivatives differential equations using the usual equivalent path integral. Since the same methodology is applied to the specific intensity spatial derivative, angular derivative and geometric sensitivity we chose to keep the same writing structure for all three parts so that all common ideas and developments appears exactly the same. We believe this choice to be coherent to facilitate the reader's understanding. Finally, these are preliminary versions of the final papers: for each parts the theory is fully described, but, although they have been implemented, the examples and algorithms sections are not always complete. This will be mentioned in the introductions of the concerned sections.