DerivedUser.php

<?php
namespace visibility;

class DerivedUser extends User
{
    public static function changeProtectedName(User $user, $name)
    {
        $user->protectedName = $name;
    }
}

Sample.php

<?php
namespace visibility;

class Sample {
    public static function main()
    {
        $user = new User();

        User::changePrivateName($user, 'João Batista Neto');
        DerivedUser::changeProtectedName($user, 'João Batista Neto');

        $user->showPrivateName(); //João Batista Neto
        $user->showProtectedName(); //João Batista Neto
    }
}

User.php

<?php
namespace visibility;

class User
{
    private $privateName;
    protected $protectedName;

    public static function changePrivateName(User $user, $name)
    {
        $user->privateName = $name;
    }

    public function showPrivateName()
    {
        echo $this->privateName, PHP_EOL;
    }

    public function showProtectedName()
    {
        echo $this->protectedName, PHP_EOL;
    }
}