Yes, all addresses are derived from one extended private key.

You can play around this tool to give you an idea:

https://iancoleman.io/bip39/

teremok

This is going to sound dumb probably, but when when people say "don't re-use addresses", they mean generate / provide a different public address for each payment / invoice you wish to accept BTC?

Is it necessary to use a different mnemonic / private key or is it sufficient to use different public addresses from the same private key by increasing the index?

bitdevs

Ask SN: accepting vanilla BTC programmatically

mikhael28

This is going to sound dumb probably, but when when people say "don't re-use addresses", they mean generate / provide a different public address for each payment / invoice you wish to accept BTC? 

Is it necessary to use a different mnemonic / private key or is it sufficient to use different public addresses from the same private key by increasing the index?