How is calculus used to analyze the stock market?

Just as the Titanic needs to slow and effectively stop before it can reverse course, so must stocks. Stocks seldom reverse on a dime and crash. There is always some period of top-building that allows astute analysts / mathematicians to follow the rate-of-change of the market. Just as a speedometer monitors the rate-of-change of a car, similar calculations can be applied to stocks. Although there is no true "instantaneous rate of change" required of a first derivative, rate-of-change calculations using stock prices of today compared with say 200 days ago can provide meaningful values to plot. I prefer using RSI (you can Wikipedia that). While prices are trending upward, expect your rate-of-change favorite to live in a "positive" position relative to its scale. When prices trend down, the rate-of-change indicator will be in a negative position -- similar to how the first derivative of a quadratic behaves. Note that the look-back periods Wall Street typically espouses is often too short to see this behavior exhibited to the extent of being obvious and able to provide you an "edge". If too short, your rate-of-change will simply oscillate between two extremes that will remind you of an EKG...

Just as parabolas have linear first derivatives, stocks on a parabolic trajectory up or down will have a pseudo-first-derivative that maintains a linear shape. By definition, this first derivative will have to break its trend before the market's actual prices do. This sneak-peek at future price action is well worth the effort of finding rate-of-change values. The 2000 peak was followed by a break of a decades-old rate-of-change trend. Most technical-analysis software packages will have numerous indicators ready-to-go, or you can often construct your own.

The shape of a rate-of-change plot will often carry hidden value too. The astute analyst will be able to interpret patterns that emerge, applying many techniques common to analyzing prices themselves (including the following).

There is also a healthy amount of algebra and geometry in the market. I use a simple formula where I square a price, divide by another price, and this quotient gives me a "target" price that I expect prices to gravitate toward. In geometry, this would appear as a "reflection" across say the X axis. If a point was 20% below the X axis, and prices moved out of a basing area, the extreme point would appear "reflected" across the axis created as prices leave the highs (lows) of the base to seek a new equilibrium. Stocks, once leaving the base, are free to climb to the target (reflected) point. Once there, traders have little incentive to hang on, expecting higher prices, so they reverse their positions. This does lead to abrupt melt-ups and melt-downs, but rarely without some re-testing of the highs(lows). That re-test would appear as a lower high(low) on a rate-of-change indicator, confirming that it's time to reverse direction. This is called a "nonconfirmation" or "failure swing", and actually shows up a fair amount during even a healthy trend. This leads many novice technicians to think that the trend is over, and they'll prematurely close out their position. The trick is find BOTH this AND a reversal target being proximate -- the ensuing reversal would be quite actionable.