Quotient Rule Derivative

The quotient rule gives the slope of one function divided by another: the top's slope times the bottom, minus the top times the bottom's slope, all over the bottom squared. Worked example: y = x²/(x + 1) at x = 1. There f = 1 and f′ = 2, g = 2 and g′ = 1, so the slope is (2 × 2 − 1 × 1)/2² = 3/4 = 0.75. A check from either side: at x = 1.001 the curve is at 0.500750 and at x = 0.999 it is at 0.499250, a rise of 0.0015 over 0.002, which is 0.75.

The classic mistake is the order on top. Addition does not care, but subtraction does: f′g − fg′ gives +0.75 here and fg′ − f′g gives −0.75, a curve going the wrong way. The rhyme students use is "low d-high minus high d-low, over the square of what's below", with the bottom function written first. The other slip is to divide the slopes, f′/g′ = 2, which is no closer. The rule's best-known job is the derivative of tan x: as sin x over cos x it gives (cos x · cos x + sin x · sin x)/cos² x, which is 1/cos² x, or sec² x. It also handles every rate that is one quantity per another: average cost per unit, a concentration as solute over a changing volume, a fuel economy as distance over fuel.