The Quotient Rule

Calculus 1 · Axiom Academy

Differentiate a fraction of two functions with one chant — "Lo d-Hi minus Hi d-Lo, over Lo squared" — then watch it work on a real example. Take two differentiable functions — a top f (the "Hi") and a bottom g (the "Lo"), with . The derivative of their quotient is built from four pieces: each function, and each function's derivative. Watch the formula assemble itself: f and g get pulled out of the fraction, paired into the two products f'g and fg' , joined by a subtraction, and set over the bottom squared. 2. The Chant: "Lo d-Hi minus Hi d-Lo, over Lo-Lo" Memorizing symbol-by-symbol is fragile. Instead, learn the rhythm. Every piece of the formula has a spoken name, and the spoken phrase is the formula read left to right. In the animation, each spoken token flips into its symbol and slides into place: Lo·d-Hi becomes , Hi·d-Lo becomes , and Lo-Lo becomes g^2 underneath the bar. The single most common quotient-rule error is swapping the numerator: writing fg' - f'g instead of f'g - fg' . Because subtraction is not commutative, that swap negates the whole answer. The animation runs both orders on and shows the result reflect across zero. Follows the chant, "Lo d-Hi minus Hi d-Lo." Gives — positive, as the rising curve demands. Terms reversed. Gives — the exact negative, so every slope comes out backwards.

This is the written version of the interactive lesson above. See the full Calculus 1 course.