Here is the script I use, in my case 6 times a day via cron. It automatically removes #backups older than 7 days. The directory is included in the directory and I save everything via rsync 6 times a day as well.
Do not put quotes for "NameDB" - "user" - "password"

#!/bin/bash

#

## we place ourselves in the directory where we want to save the databases

#

cd /var/www/exemple/backup/bd/

for i in "NameDB"; do

## Backup of databases in .sql files (with date as name)

mysqldump -u "user" -p"password" $i > ${i}_`date +"%d%m%Y"`_`date +"%H"`h`date +"%M"`m.sql


## Compression of exports in tar.bz2 (the best compression rate)

tar jcf ${i}_`date +"%d%m%Y"`_`date +"%H"`h`date +"%M"`m.sql.tar.bz2 ${i}_`date +"%d%m%Y"`_`date +"%H"`h`date +"%M"`m.sql


## Elimination of uncompressed exports

rm ${i}_`date +"%d%m%Y"`_`date +"%H"`h`date +"%M"`m.sql

## Deletes backups older than 7 days


find /var/www/exemple/backup/bd/ -type f -mtime +7 -delete

## Sent an email confirmation

echo "Sauvegarde BDD vers var/www/exemple/backup/bd et nettoyage effectue avec succes !" | mail -s "Save BDD Meetsee" -a "From: sender@yoursite.com" recipient@yoursite.com

done