If you understand this derivation deeply, you understand the basics of nearly every aspect of finance, including arbitrage, risk management, valuation, hedging, Itō's lemma, short selling, mergers, market microstructure, portfolio management, yield curves, hedge funds, behavioral finance, and more. And all this in just five short equations.
Parameter Greeks like vega and rho are fundamentally different from sensitivity Greeks like delta and theta. They are far more important. Sensitivity Greeks are simultaneous outputs from the model. They are supplemental information to go along with the model price. If you think of the model as a person, then the model price is its body, and the model sensitivity Greeks are its accentuating makeup and jewelry and clothes. The model is proud to calculate and display them for you. The underlying will bounce around; time will pass; the model knows this and happily exhibits your sensitivities to those events. But parameter Greeks are an embarrassment to the model. They are the cracks beneath the makeup, the scars beneath the clothes, the hollowness inside the body. They expose the fact that the model is false.
One of the most common mistakes that even highly experienced practitioners make is to act as if the assumptions of Black-Scholes (lognormal, continuous distribution of returns, no transactions costs, etc.) mean that we can always arbitrarily assume the underlying grows at the riskfree rate r instead of a subjective guess as to its real drift μ. But this is not quite accurate. The insight from the Black-Scholes PDE is that the price of a hedged derivative does not depend on the drift of the underlying. The price of an unhedged derivative, for example, a naked long call, most certainly does depend on the drift of the underlying.
Let's say you are naked long an at-the-money one-year call on Apple, and you will never hedge. And suppose Apple has very low volatility. Then the only way you will profit is if Apple's drift is positive; suppose Apple has very low volatility. Then the only way you will profit is if Apple's drift is positive…if it drifts down, your option expires worthless. But if you hedge the option with Apple shares, then you no longer care what the drift is. You only make money on a long option if volatility is higher than the initial price of the option predicted.