Is a symmetric encryption key derivable using the input and the output?
For example when I encrypt message A with key K and it outputs B, can somehow A and B be put together, so it will output K?


Except for encryption messages where you are specifically restricted to use a specifc key K only once (e.g. OTP), then no, it is infeasible to recover K from A and B.

This is a fairly fundamental requirement on encryption methods (known as "Known Plaintext"); that is, given a plaintext message and the corresponding ciphertext (or more generally, a huge number of messages and ciphertexts, all encrypted with the same key), that is still no help in decrypting another ciphertext (also encrypted with the same key) - being able to recover the key from a plaintext and ciphertext would violate this.

Actually, we put even tougher requirements on encryption methods.  We allow the attacker to pick the messages to be encrypted ("Chosen Plaintext"), and also allow the attacker to submit his own ciphertexts (and see how they are decrypted), "Chosen Ciphertext".

An encryption method is generally not considered secure unless it is secure against all these attack scenarios.