I was trying to update from Drupal 7.34 to the current (7.39) version. I keep getting the following error on the status report page after updating.

Fatal error: Class 'DatabaseTasks_mysql' not found in modules/system/system.install

The solution that worked for me was emptying the registry and registry_file tables, then running update.php.

Update.php didn't find any database changes, but it reset the registry. 

If you have installed phpmyadmin, go to phpmyadmin and then select the drupal database. Select the tables named, registry and registry_file and choose the action as Empty found at the bottom of the page. This action will truncate the tables. After that run update.php to rebuild the registry and then clear the cache.