Why is delta a *partial* derivative and not just an ordinary derivative?

Delta is a partial derivative because the option price $V$ depends on more than one variable. The notation $\partial V / \partial S$ (curly d) explicitly means "rate of change of $V$ with respect to $S$, holding everything else constant."

The six inputs to BSM:

1. $S$ — underlying stock price
2. $K$ — strike (constant by contract)
3. $r$ — risk-free rate
4. $q$ — dividend yield
5. $\sigma$ — volatility
6. $T$ — time to expiry

If we wrote $dV/dS$ as an ordinary derivative, we would implicitly require $S$ to be the only variable changing — which is never true in practice. In reality, the moment the stock moves, time has also advanced (theta), and possibly implied vol has shifted (vega), and possibly rates have changed (rho).

Each Greek isolates one variable:

Each Greek is a partial derivative because each one tells us the response to ONE variable while everything else is frozen.

The practical consequence:

When you delta-hedge, you neutralise one specific risk — stock-price risk — without affecting your exposure to volatility, time, or rates. If delta were the ordinary derivative dV/dS, you would also be hedging vega and theta simultaneously, and you would have no way to separate "I am hedged for stock moves" from "I am hedged for vol changes."

Total derivative interpretation:

If you want the total change in $V$ when multiple variables move simultaneously, you can write:

This is a Taylor expansion using the Greeks. Each Greek is one partial derivative coefficient. The product of the right Greek and the right increment gives the contribution from that risk factor.

So delta being a partial derivative is what lets us decompose option risk into independent buckets. That decomposition is the entire business model of an options desk.