Library / Foundations: Maths / FMA-24

Foundations: Maths

Product rule

Derivative of a product of two functions.

Ref: FMA-24
Rev: v1
Field: Foundations: Maths
Result: (uv)′

(u v)' = u' v + u v'

Primes denote derivatives with respect to x. u, v are the function values and u′, v′ their derivatives at the same x.

When two changing quantities are multiplied, the rate of change of the product has one contribution from each.

Calculator

Solve for

(uv)′Derivative of uvComputed

110

u′u′

0.55

110

v′v′

0.58

Product uv12
u′v term8

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.

Derivative of uv () current value 23

uuKey driver

1358

v′v′

9.532

u′u′

1735

Never just multiply the two derivatives; each function takes a turn at being differentiated.
Power = force × velocity gives dP/dt from the changes in both.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
u u	1 to 10	13	58	45	+10%	+6.52 %
v′ v′	0.5 to 8	9.5	32	22.5	+10%	+6.52 %
u′ u′	0.5 to 5	17	35	18	+10%	+3.48 %
v v	1 to 10	17	35	18	+10%	+3.48 %

Sweep of v′

v′	Derivative of uv
0.5	9.5
1.438	12.31
2.375	15.13
3.313	17.94
4.25	20.75
5.188	23.56
6.125	26.38
7.063	29.19
8	32

Sweep one variable

Vary

Derivative of uv ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

(uv)′

Derivative of uv

Result ·

What the equation gives you. Everything else feeds it.

u

· typical 1 to 10

Value of the first function. Multiplies v′.

u′

· typical 0.5 to 5

Derivative of the first function. Multiplies v.

v

· typical 1 to 10

Value of the second function. Multiplies u′.

v′

· typical 0.5 to 8

Derivative of the second function. Multiplies u.

Worked example with the values above

First term: u′v = 2 × 4 = 8.
Second term: uv′ = 3 × 5 = 15.
(uv)′ = 23.

Assumptions and limits

u and v are differentiable at the point.
Derivatives are taken with respect to the same variable.

Common mistakes

Using (uv)′ = u′v′.
Dropping one of the two terms.

Statistics and uncertainty

If u and v are measured with their own relative errors, both terms carry error; the combined error depends on the size of each term.

Where it is used

Rate of change of power, momentum and area.
Differentiating products in dynamics.

Maths you need

Differentiation: rate of change
Functions: values and derivatives

Related formulas

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