1. Financial derivatives have no relationship to derivatives in calculus. (The two concepts are just coincidental namesakes.) A financial derivative is called a "derivative" because the price of a financial derivatives is _derived_ from the price of an underlying something. However, the price of the underlying asset is just one of many inputs into the price of a derivative on that option, so nobody is taking a "first derivative with respect to price".
2. It would be impossible to make a billion dollar bet on a penny stock using "plain vanilla" (regular) derivatives like options, because stock options are not offered on thinly traded penny stocks (because nobody cares about the stock, so nobody will care about the stock option, so an exchange won't invest in offering that stock option). The only way you could make a billion dollar bet on a penny stock is if you found an large institution or billionaire to make a customized bet with you about that stock (or a coin toss). However, there is almost no liquidity for a non-exchange-traded option of this kind.
3. A stock option and its underlying stock have an intimate relationship. An options trader will often buy or short a stock to hedge the risk s/he has from owning the stock option (and vice versa). An options/derivative trader definitely has to worry about how "unwinding a position" in one market will affect the other market.