Quotient rule
The derivative of u over v is (u'v - uv') divided by v^2. The order of the top terms matters.
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Analogy
Low d-high minus high d-low, over low squared: a rhyme to keep the order straight.
Example
u = x, u' = 1. v = x + 1, v' = 1.Top: 1(x + 1) - x(1) = 1.Result: 1 / (x + 1)^2.
In practice
Use it for fractions with x on top and bottom, unless rewriting as a product is easier.
Practice
Flashcards
- State the quotient rule.(u/v)' = (u'v - uv') / v^2.
Questions people ask
Do I have to use the quotient rule for every fraction?
No. If the bottom is just a number, use the power rule. If it is easier, rewrite u over v as u times v to the power -1 and use the product and chain rules. Both ways give the same answer, so choose the less messy one.
Why is there a minus sign in the quotient rule?
Increasing the denominator makes the fraction smaller, so its change counts against the result. That is the minus in u prime v minus u v prime. Swapping the order of the two terms would flip the sign and give a wrong answer.