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If I use ssh like this ssh -i id_rsa it means im passing the private key to ssh. Does ssh now then compute the public key from it and then send it to the server? Then the server would look if the public key exists in the authorized_keys file
Or does it look for the public key in ~/.ssh/id_rsa.pub? What if the client doesnt have the public key but only the private key?
It used to read .pub and still can.
In past years OpenSSH (specifically ssh-keygen) stored privatekeys in OpenSSL-defined formats, which cryptographically allow derivation of the publickey (as Steffen commented) but only after decrypting, which requires you to enter the password. To reduce wasted effort and time, if you specified ssh -i file it looked for file.pub and if found used the (clear) publickey to 'probe' the server to determine if that keypair is suitable (i.e. present in authorized_keys, or an equivalent for other software) before prompting you for the password and actually using the (private)key to authenticate.
Beginning with OpenSSH 6.5 in 2014 if you specify -o or use Ed25519, and with OpenSSH 7.8 in 2018 by default (for all keytypes), ssh-keygen uses 'new format' files, which contains the publickey in clear and only the privatekey (normally but not always) encrypted, so ssh can do the probe logic without a .pub file.
Also if you have the key loaded in an agent process, the password is only specified at the time the key is loaded and both the privatekey and publickey are subsequently available without a password. This option has existed for a long time, but in past years it had to be invoked/configured manually and many people didn't know and/or bother. In recent years it has become fairly common (but not universal) for systems or environments (e.g. Linux distros) to set up an SSH agent for you, either OpenSSH's own ssh-agent or an alternative like gpg-agent or gnome-keyring.
Thus nowadays ssh usually doesn't need the .pub file. But if you use legacy-format encrypted keyfile(s) not loaded in an agent and don't have corresponding .pub file(s), you may be forced to enter password(s) for key(s) that then is(are) skipped because the server doesn't accept it(them).
