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I know that you can generate a new key pair with a command like.

openssl ecparam -genkey -name secp256k1

And the resulting output will contain a new private key with the encoded public key.

Is it possible for openssl to generate the public key straight from an existing private key only?

Just to be clear I mean Bitcoin Public Key and not Public Address

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Yes, openssl shows a public key corresponding to a private key that is input, e.g.

$ openssl ec -in priv_key.openssl -text -conv_form compressed

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