VG3
kalkulus
Quiz
What is the quotient rule?
The quotient rule is used when differentiating a fraction of two functions.
When a function is a fraction of two other functions we use the quotient rule.
[u/v]' = (u'v - uv') / v²
Numerator: derivative-of-top times bottom, minus top times derivative-of-bottom.
Denominator: the bottom squared.
Memory tip: "low-D-high minus high-D-low, over low squared"
(low = denominator, high = numerator, D = differentiate)
(low = denominator, high = numerator, D = differentiate)
Examples
Example 1: Differentiate h(x) = sin(x) / x
u = sin(x) → u' = cos(x)
v = x → v' = 1
h'(x) = (cos(x) · x - sin(x) · 1) / x²
= (x·cos(x) - sin(x)) / x²
v = x → v' = 1
h'(x) = (cos(x) · x - sin(x) · 1) / x²
= (x·cos(x) - sin(x)) / x²
Common mistake: Watch the order! It is u'v - uv', NOT uv' - u'v. The sign matters!
In mathematics the art of proposing a question must be held of higher value than solving it.
— Georg Cantor (1845–1918)
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What is the quotient rule?
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