Library / Foundations: Maths / FMA-25
Foundations: Maths
Quotient rule
Derivative of one function divided by another.
- Ref
- FMA-25
- Rev
- v1
- Field
- Foundations: Maths
- Result
- (u/v)′
Primes denote derivatives with respect to x. v must be non-zero.
The rate of change of a ratio depends on how fast the top changes against the bottom, divided by the square of the bottom.
Calculator
- Ratio u/v0.75
- Numerator u′v − uv′17
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The order in the numerator matters: u′v first, then subtract uv′.
- You can instead write u/v as u·v⁻¹ and use the product and chain rules.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| u′ u′ | 2 to 8 | 0.3125 | 1.813 | 1.5 | +10% | +11.8 % |
| v v | 2 to 8 | 0.5781 | 1.75 | 1.172 | +10% | −7.63 % |
| v′ v′ | 0.5 to 3 | 0.6875 | 1.156 | 0.4688 | +10% | −1.76 % |
| u u | 1 to 6 | 0.875 | 1.188 | 0.3125 | +10% | −1.76 % |
Sweep of u′
| u′ | Derivative of u/v |
|---|---|
| 2 | 0.3125 |
| 2.75 | 0.5 |
| 3.5 | 0.6875 |
| 4.25 | 0.875 |
| 5 | 1.063 |
| 5.75 | 1.25 |
| 6.5 | 1.438 |
| 7.25 | 1.625 |
| 8 | 1.813 |
Sweep one variable
Variables and how each one contributes
Derivative of u/v
What the equation gives you. Everything else feeds it.
u
Value of the numerator function. Lowers the derivative when v′ is positive.
u′
Derivative of the numerator. Raises the derivative.
v
Value of the denominator function. Appears squared in the denominator.
v′
Derivative of the denominator. Lowers the derivative.
Worked example with the values above
- Numerator: u′v − uv′ = 20 − 3 = 17.
- Denominator: v² = 16.
- (u/v)′ = 1.0625.
Assumptions and limits
- v ≠ 0 at the point.
- Both functions are differentiable.
Common mistakes
- Swapping the two terms in the numerator.
- Forgetting to square v.
Statistics and uncertainty
When v is small the derivative becomes large and very sensitive to errors in v.
Where it is used
- Rates of change of efficiency and other ratios.
- Differentiating tan x = sin x / cos x.
Maths you need
- Differentiation: rates of change
- Fractions: division of functions