The Quotient Rule | "Low D-High Minus High D-Low"

Can you just divide the derivatives? Test it on x²/x = x: the real derivative is 1, but 2x/1 = 2x. The guess fails. In this video you'll learn the real quotient rule with the classic rhyme — "low d-high minus high d-low, square the bottom and away we go" — see WHY it works (it's the product rule in disguise, and a limit proof), learn when NOT to use it, and practice with 21 worked examples, from warm-ups to AP-style tangent-line and table problems. Calculus 1 · AP Calculus AB/BC · A-Level · IB. In this video you'll learn: ✔ The quotient rule: (f/g)′ = (g·f′ − f·g′)/g² ("low d-high minus high d-low, over low squared") ✔ Why it works: f/g = f · g⁻¹ with the product rule (plus a "trust me" from the chain rule, proved in DC-07) ✔ The formal proof with the limit definition (the subtract-and-add trick) ✔ When NOT to use it: rewrite as a power (3/x⁴ = 3x⁻⁴), split ((x² + x)/x = x + 1), or pull out a constant ✔ A 4-step routine with a pit board: high f, low g, f′, g′ ✔ The traps: the ORDER of the top (g·f′ − f·g′, not f·g′ − g·f′) and forgetting to square the bottom ✔ 15 Rule Gym examples + 4 Race Ex + 2 Error Detective cases: x/(x + 1) · (2x + 3)/(x − 4) · 6/√x · (x³ + 2x)/5 · 1/(x² + 1) · (x² − 1)/(x² + 1) · sin x/x · eˣ/x² · (2x³ − x + 4)/x · (x² + 3)/(x − 1) · √x/(x + 1) · tan x = sin x/cos x (→ sec²x) · AP tangent line to y = (x + 3)/(x − 1) at x = 3 · AP table problem h = f/g, h′(2) · x eˣ/(x + 1) Race Ex: x/(x − 2) · (x² + 4)/x · x²/eˣ · (3x − 1)/(x + 2) · Error Detective: x²/(x + 1) · x³/(x² + 1) (For now we trust (sin x)′ = cos x, (cos x)′ = −sin x and (eˣ)′ = eˣ — they get their own videos: DC-09 and DC-10.) ⏱️ CHAPTERS 0:00 Is (f/g)′ = f′/g′? 0:55 The quotient rule + the song 2:07 Why it works: the product rule in disguise 3:19 The limit proof (subtract and add) 4:41 When NOT to use the quotient rule 5:42 The routine + your toolbox (sin, cos, eˣ) 6:43 Rule Gym: 15 worked examples 15:50 Race Ex: 4 ten-second races 18:20 Error Detective 19:38 The traps: the order and the square 20:36 3-card recap 20:59 Wow fact: negative exponents for free ◀️ Previous: DC-05 · The Product Rule | The Growing Rectangle ▶️ Next: DC-07 · The Chain Rule | Gears Inside Gears 📚 Full playlist: The Speedway · Differential Calculus 💬 Tell me in the comments: what is the derivative of x/(x² + 1)? (Hint: sing it!) 👍 If this helped, hit like and subscribe — new visual math every week. 🇪🇸 ¿Prefieres en español? → @AlgebrayTrigonometria #calculus #derivatives #APcalculus #calculus1