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Delta, gamma, theta, vega and rho are five different sensitivities of one option price, and most retail traders watch exactly one of them. That is why a stock can move your way and your call still loses money.
A call is a bet that something will go up. A put is a bet that something will go down. That’s most of what you need to know, but the small print is where people lose money.
The quotient rule is the calculus shortcut for taking the derivative of a fraction where both the top and the bottom depend on x. It's mostly memorization, but understanding why the formula has the form it does makes the rule much harder to forget.
