How to publicly inherit from a base class but make some of public methods from the base class private in the derived class?

Section 11.3 of the C++ '03 standard describes this ability:

11.3 Access declarations
The access of a member of a base class can be changed in the derived class by mentioning its qualified-id in the derived class declaration. Such mention is called an access declaration. The effect of an access declaration qualified-id ; is defined to be equivalent to the declaration using qualified-id

So there are 2 ways you can do it.

Note: As of ISO C++ '11, access-declarations (Base::bar;) are prohibited as noted in the comments. A using-declaration (using Base::bar;) should be used instead.

1) You can use public inheritance and then make bar private:

class Base {
public:
    void foo(){}
    void bar(){}
};

class Derived : public Base {
private:
    using Base::bar;
};

2) You can use private inheritance and then make foo public:

class Base {
public:
    void foo(){}
    void bar(){}
};

class Derived : private Base {
public:
    using Base::foo;
};

Note: If you have a pointer or reference of type Base which contains an object of type Derived then the user will still be able to call the member.

There is really no way to do what you want because if you derive publicly from Base, a user of the class will always be able to:

Derived d;
Base& b = d;
b.bar();

It isn't "correct or natural" to make public base class functions inaccessible in a class derived publicy from the base class; instead, the base class interface should be refactored such that those functions are not public or are split into a separate class.

First, you have to understand what you want to do from the OOP perspective. There are two completely different types of inheritance:

1. Inheritance of Interface. It is when you do implement in Java or other languages that have interfaces as standalone entities, it is also happens when you publically inherit from an empty abstract class in C++. Here you don't care about code at all but want to tell your compiler and everyone using you base/derived classes that this derived class is a special kind of you base class, it has all the properties of base class, it behaves exactly as base class does to the extent visible by the user of it, and can be used instead of the base class in any algorithms.

2. Inheritance of the code. You have a piece of code in base class that you want to reuse in your derived class. Base class and derived class does not have to be related in any way, you just want to reuse code and that is it.

Public inheritance in C++ is the mix of both kinds, you get interface inheritance and you get code inheritance as well. Private inheritance is a different kind of beasts, you get only code inheritance, users of your derived class can not use it instead of base class and from the user perspective base and derived classes have no relation what-so-ever.

struct Base {};
struct PublicDerived : public Base {};
struct PrivateDerived: private Base {};

Base * base; PublicDerived * public_derived; PrivateDerived * private_derived;

base = public_derived; //good
base = private_derived; //compilation error.

Since you want to change the interface, you shouldn't go with public inheritance, by changing the interface you effectively saying that those two classes have different behavior and can not be used interchangeably. So what you really want is privately inherit and then make all the methods you want public and not the other way around.

According to the Liskov Substitution Principle, public inheritance should model "is-a." What you are saying with Derived publicly inheriting from Base is that wherever Base object is required, an object of type Derived will do.

If you requre Derived to hide some operations that are available for Base, then what you are modeling is something other than "is-a," and public inheritance is not the correct tool.

What you want is either private inheritance or composition, as other answers have detailed.