Key takeaways
- Quotient rule: (f ÷ g)′ = (f′g − fg′) ÷ g².
- Memory trick: low d-high minus high d-low, over low squared.
- Order matters: swapping the two terms on top flips the sign.
- Don’t forget to square the bottom, and never just divide the derivatives.
In this guide
What is the quotient rule?
A derivative is the rate of change, or the slope of a graph at a point. When your function is a fraction with x on the top and the bottom, both parts change at once. The quotient rule combines those two changes correctly.
Why does the quotient rule matter?
Lots of useful quantities are ratios: average cost per item, speed = distance ÷ time, the concentration of medicine in the blood. The quotient rule tells you how fast a ratio is changing.
It also unlocks new derivatives. The derivative of tan x comes straight from writing it as sin x ÷ cos x.
When the bottom of a fraction grows, the whole fraction shrinks. That shrinking effect is exactly what the minus sign in the quotient rule measures.
The formula
| Symbol | What it means | Example (x ÷ (x + 1)) |
|---|---|---|
| f | The top (numerator), “high” | x |
| f′ | Derivative of the top, “d-high” | 1 |
| g | The bottom (denominator), “low” | x + 1 |
| g′ | Derivative of the bottom, “d-low” | 1 |
| g² | The bottom squared | (x + 1)² |
This calculator uses f(x) = xᵃ and g(x) = xᵇ and gives the slope at your chosen x. You can check it with the power rule, since xᵃ ÷ xᵇ = xᵃ⁻ᵇ.
How to use the quotient rule step by step
- 1Write down four things. f, g, f′ and g′.
- 2Low d-high. Work out f′ × g.
- 3Minus high d-low. Work out f × g′ and subtract it.
- 4Over low squared. Divide by g².
- 5Simplify. Cancel anything you can.
Now try it yourself. Enter the top power, the bottom power and x in the calculator below to see the top and bottom worked out separately.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Forgetting to square the bottom. The denominator is g², not g.
- Dividing the derivatives. (f/g)′ is not f′/g′. For x⁴ ÷ x² at x = 3, the true slope is 6, but
f′ ÷ g′ = 4x³ ÷ 2x = 2x² = 18. - Using x = 0 when the bottom is a power of x. The bottom becomes zero, so the calculator shows an error.
Tips and tricks
- Simplify first when you can. x⁴ ÷ x² is x², so the slope is just 2x. Use the quotient rule when it won’t simplify, like x ÷ (x + 1).
- Or avoid it. Rewrite
f ÷ gasf × g⁻¹and use the product rule with the chain rule. You get the same answer. - Famous result:
tan x = sin x ÷ cos xgives(cos²x + sin²x) ÷ cos²x = 1/cos²x = sec²x.
Where you'll use it in real life
- Business: average cost = total cost ÷ number of items. Its derivative shows when making more gets cheaper per item.
- Medicine: drug concentration formulas are often fractions in time.
- Physics: rates like speed and density are ratios.
- Trigonometry: the derivatives of tan, sec and cot all come from the quotient rule.
Quick summary
To differentiate a fraction, use (f′g − fg′) ÷ g²: low d-high minus high d-low, over low squared. Keep the order on top, and remember to square the bottom. Use the calculator above to check xᵃ ÷ xᵇ at any point, with every step shown.
Worked example
A worksheet asks for the slope of y = x⁴ ÷ x² at x = 3 using the quotient rule, so you can check it against the simpler form x². The top has power a = 4 and the bottom has power b = 2.
- 01f(x) = x^4, g(x) = x^2
- 02f'(x) = 4x^3, g'(x) = 2x^1
- 03Quotient Rule: (f'g − fg') / g²
- 04At x = 3: f = 81, g = 9, f' = 108, g' = 6
- 05Numerator = 108 × 9 − 81 × 6 = 486
- 06Denominator = (9)² = 81
- 07Result = 486 / 81 = 6
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the easiest way to remember the quotient rule?
Say “low d-high minus high d-low, over low squared”. Low is the bottom function, high is the top, and d-high means the derivative of the top.
Can I avoid the quotient rule?
Sometimes. You can rewrite f/g as f × g⁻¹ and use the product rule with the chain rule. You get the same answer.
Why is there a minus sign in the quotient rule?
When the bottom grows, the fraction gets smaller. The minus sign accounts for that shrinking effect.
What is the derivative of 1/x using the quotient rule?
f = 1, f′ = 0, g = x, g′ = 1. So (0 × x − 1 × 1) ÷ x² = −1/x², the same as the power rule gives.
