Product Rule Derivative

The product rule gives the slope of two functions multiplied together: the slope of the first times the second, plus the first times the slope of the second. Worked example: y = x²(x³ + 1) at x = 2. There f = 4 and f′ = 2x = 4, while g = 9 and g′ = 3x² = 12, so the slope is 4 × 9 + 4 × 12 = 36 + 48 = 84. Expanding first agrees: x²(x³ + 1) = x⁵ + x², whose slope at 2 is 5 × 2⁴ + 2 × 2 = 80 + 4 = 84. The page asks for the four numbers rather than the functions because that is where the rule is actually applied: once f, f′, g and g′ are known at the point, what is left is arithmetic.

The classic mistake is to multiply the two slopes, f′ × g′, which here gives 4 × 12 = 48 and is wrong whenever both factors are changing. A rectangle with sides f and g shows why. Grow both sides a little and the new area arrives as two strips, one g long and Δf wide, one f long and Δg wide; the only place a product of two changes appears is the tiny corner square Δf·Δg, and it vanishes in the limit. Leibniz printed the rule in 1684, in the first published paper on calculus. It turns up wherever a quantity is a product of two changing things: revenue is price times quantity sold, electrical power is voltage times current, and the momentum of a rocket is a mass that is falling multiplied by a speed that is rising.