Hierarchical Deterministic (HD) wallets are described in BIP-32. There are a number of technical aspects to this, so for the simplicity of this answer, I'll only talk about deriving a child private key from a parent private key, using non-hardened derivation.
To keep the syntax similar to your screenshot, let
y be the parent private key, so that
g^y is the parent public key. In BIP-32 HD, we don't just use
y though, we also need the chain code for
y. They call it a random value, but in practice it is computed in a deterministic way, so that it can be retrieved later. Let's call this
k, as the purpose in the screenshot seems to be similar.
To derive child
i, we compute (simplified):
I = HMAC-SHA512(k, g^y || i).
k is used as a secret to compute the HMAC function. In the screenshot they simply hashed the two together, which isn't best practice, but may be simpler to understand. They also left out
g^y, so let's ignore that too.
I we computed is 512 bits long. Split it in half, and call the left 256 bits
L and the right 256 bits
R is the child chain code (which you can ignore if you like), used to derive children of this child and continue the chain.
- The child secret key is
y + L.
Because your screenshot simply just computes a hash, you can think of it like directly computing
L here without computing
R. It is safe to say that
x is a typo though. HD derivation only needs the chain code
k and private key
y to compute the child private keys.