#1 - There is no way you will be able to express much useful actuarial notation without learning some LaTeX. Even WITH LaTeX, it will be a struggle, but we can get through this one.
#2 - Normally, is defined only annually. Its derivative tends to be zero unless you take some time to define it between year ends. If you want to talk differences, , that's a different story. If you have a continuous , you'll have to state that.
#3 - You probably mean , where s(t) is the survival function for the year of interest. It is continuous and should have a derivative, except maybe at the year ends.
Where does that leave us?