Wallet Derivation Path Question

I have a really basic cryptography question that I can't seem to find a straight answer for on Google.

Let's say I create another wallet on my hardware device by changing the derivation path. Is there a way for someone to figure out that the NEW wallet comes from the same private key? I am not asking whether they can figure out the private key. I know that answer to that is no. I am asking whether the relationship between two wallet addresses from the same private key is also a secret.