Sizu Riatto
Calculus · Chapter 11

Chain of Rates

Rates multiply along a chain, like gears: dollars per hour = dollars per gallon × gallons per mile × miles per hour. That's the chain rule. And when price and sales both change, revenue changes along both sides of a rectangle: the product rule.

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Opens on Chapter 11’s example: a drive at $3.60 a gallon, 30 miles a gallon and 60 mph; a shop at $40 rising $2 a month with sales of 300 falling 10.

1. A chain of rates
2. The chain rule on (ax + b)ⁿ
3. Price and sales both changing
Enter your numbers, then press Work it out.

How it works

Write each rate as a fraction of units and multiply: a unit on the bottom of one link cancels the same unit on top of the next, so what's left tells you what the answer measures. The chain rule is the same idea for formulas: the rate of the outside × the rate of the inside, so d/dx (3x + 1)² = 2(3x + 1) × 3. The product rule is a growing rectangle: when price p and sales q both change, revenue pq changes by (change in p) × q plus p × (change in q). The tiny corner piece, (change in p) × (change in q), vanishes for small changes but shows up over a whole month.

From the book, Chapter 11. 80 × 2.5 × 7 gives 1,400 feet a minute, about 15.9 mph. At $3.60 a gallon and 30 mpg, 60 mph costs $7.20 an hour. The patch (3x + 1)² grows at 78 square feet per foot at x = 4. The shop: price’s part $600, sales’ part −$400, revenue rising at $200 a month; the real month goes from $12,000 to $12,180, up $180, the $20 difference being the corner piece. Every chapter and its calculator →

Numbers can be typed as 1,450, $38, 0.40, 12%, 1/6 or 2 3/8. Functions can be typed as 16t², 6 + 48t − 16t^2, 1000e^(0.05t) or √x; answers are rounded half up, the way the book rounds. Everything stays in your browser: no sign-up, nothing stored or sent.