Mistake Master
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The Quotient Rule AB & BC

The quotient rule is $\frac{d}{dx}\left[\frac{u}{v}\right] = \frac{u'v - uv'}{v^2}$: the numerator's derivative leads, the two terms are subtracted in that order, and the whole denominator is squared. Because subtraction is not commutative, swapping the terms produces exactly the negative of the correct answer. A constant denominator or a single power underneath should be rewritten instead, and a denominator with two terms has to go through the rule.

The errors are order, sign, and shape. The two numerator terms get reversed, the minus fails to distribute across a multi-term numerator, or the denominator is left as $v$ rather than $v^2$. Behind those sits the same instinct that the product rule has to defeat: differentiating the top and the bottom separately and dividing the results, which fails on the very first example anyone tries.

(u/v)′ = u′v - uv′ v² the TOP is differentiated first then the bottom, and it is subtracted the whole denominator, squared swap the two terms and you get the NEGATIVE of the right answer a quotient rule answer whose denominator is not squared used the wrong rule and u′/v′ is not a rule: for x²/x it gives 2x where the true derivative is 1
The only structural difference from the product rule is a minus sign, and it is the one that makes order matter.
f(x) = (2x + 1) / (x - 3) u = 2x + 1 u′ = 2 v = x - 3 v′ = 1 [ 2(x - 3) - (2x + 1)(1) ] / (x - 3)² = [ 2x - 6 - 2x - 1 ] / (x - 3)² = -7 / (x - 3)² the minus distributes across BOTH terms negative everywhere, and f is indeed decreasing
Brackets around the second product, then expand. That one line is where the sign is usually lost.

The work

3 ways in · any order
Lesson
The Quotient Rule ›

Fixes the order of the quotient rule's two numerator terms and the squared denominator, drills distributing the minus sign across a multi-term numerator, and marks the quotients that should be rewritten rather than run through the rule.

Skill check · 10 scenarios
Open lesson →
Diagnostic
10-item topic check ›

Ten items spanning the two failure modes of this topic: reversing the numerator's two terms, losing the minus sign or failing to square the denominator, and dividing the derivative of the top by the derivative of the bottom.

Not yet available · 10 items
Coming soon
Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.

Take the diagnostic to identify your misconceptions
Diagnostic first