No, you cannot access derived_int because derived_int is part of Derived, while basepointer is a pointer to Base.
You can do it the other way round though:
Derived* derivedpointer = new Derived;
derivedpointer->base_int; // You can access this just fine
Derived classes inherit the members of the base class, not the other way around.
However, if your basepointer was pointing to an instance of Derived then you could access it through a cast:
Base* basepointer = new Derived;
static_cast<Derived*>(basepointer)->derived_int; // Can now access, because we have a derived pointer
Note that you’ll need to change your inheritance to public first:
class Derived : public Base