How to understand the ill-posedness of Volterra equation of first kind and well-posedness of second kind?

Question

Consider a Volterra equation of first kind as following: $$ f(t) = \int_0^t K(t,s) x(s) ds $$ We can change it to second kind by taking derivative of both sides: $$ \tilde f(t) = x(t) + \int_0^t \tilde K(t,s) x(s) ds $$ The above two equations are equivalent, which means they have same unique solution. And as we know, the first kind Volterra equation is ill-posed because the does not continuously depend on data, but the second kind equation is well-posed. So in practice, people often change it to second kind and numerical solve it. So we find there are two equivalent problems: one is ill-posed and the other is well-posed. I was wondering if it is possible that the two equivalent problems have different properties. And what happened in the transform from first kind to second kind and make it become well-posed? Thanks a lot.