After I understand the delta derivation from BSM, what other Greeks can I derive using the same technique?

All five major Greeks can be derived from the BSM formula using the same partial-derivative machinery, but they vary in algebraic difficulty. Here is the full taxonomy.

Delta ($\partial c / \partial S$) — easy: the derivation we just walked through. Result: $\Delta_{\text{call}} = e^{-qT} N(d_1)$.

Gamma ($\partial^2 c / \partial S^2$) — easy if you have delta: differentiate delta one more time. Result:

Same for calls AND puts (gamma is symmetric). Always positive for long options.

Vega ($\partial c / \partial \sigma$) — medium: differentiate the call formula with respect to $\sigma$. Both $d_1$ and $d_2$ depend on $\sigma$, so chain rule on both. After applying the magical $N'(d_1) S = N'(d_2) K e^{(r-q)T}$ identity, you get:

Same for calls AND puts. Always positive (more vol means more option value).

Theta ($\partial c / \partial T$) — hard: the most algebraically painful Greek. Multiple terms because $d_1$ and $d_2$ both depend on $T$, AND the discount factors $e^{-rT}$ and $e^{-qT}$ depend on $T$ directly. After heroic algebra:

Negative for long calls (time decay hurts). Slightly different signs for puts.

Rho ($\partial c / \partial r$) — easy: differentiate with respect to $r$. The only term that depends on $r$ is $K e^{-rT} N(d_2)$, but $d_2$ also depends on $r$. After applying the identity again:

Positive for calls (higher rates make calls more valuable for non-dividend stocks because the forward $F = S e^{(r-q)T}$ rises).

Difficulty ranking (easiest to hardest):

1. Delta — one application of chain rule + the $N'$ identity
2. Gamma — differentiate delta once more, comes from chain rule on $N(d_1)$
3. Vega — chain rule on $\sigma$, identity cancels most terms
4. Rho — only one term depends on $r$, identity gives clean cancellation
5. Theta — multiple time dependencies, multiple terms survive

The recurring trick:

Every Greek derivation uses the same algebraic identity $S e^{-qT} N'(d_1) = K e^{-rT} N'(d_2)$ to cancel $N'$ terms. Once you have proven that identity once, all five Greeks fall out of the same machinery. This is why BSM is so beloved by quants — the math has internal symmetry that makes every related calculation tractable.

What CFA Level II expects:

You should know the closed forms for $\Delta$, $\Gamma$, $\nu$, $\Theta$, $\rho$ and be able to apply them in numerical questions. You do NOT need to reproduce the derivations on the exam. Focus your study on plugging numbers in, interpreting signs (Greeks symbols), and understanding what each Greek tells you about risk.