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Is it possible to derive a new private key from an existing private key and retain the same bitcoin balance without any transactions?

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No. A private key corresponds to one public key (and vice versa), and that corresponds to one address (and vice versa). Having a new private key (regardless of where it comes from, so it could be derived from a different private key) would mean that you have a new public key and a new address. Thus you cannot spend any funds associated with the first address with the new private key because it is not associated with that address.

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    Technically, you can derive two addresses from a single private key. Bitcoin supports both compressed and uncompressed public key formats, which can both be used to generate addresses. Don't be fooled, however. It's still the same public key, just written differently. See bitcoin.stackexchange.com/a/3062/24974 – Jestin Dec 23 '17 at 6:09

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