Step by Step
1
The rule itself
For (f/g)', the derivative is (g·f' − f·g') / g² — the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all over the denominator squared.
2
The classic memory phrase
"Low D-High minus High D-Low, square the bottom and away we go." Low = the denominator (g), High = the numerator (f), D = derivative. This phrase locks in the correct order of subtraction.
3
Order matters — subtraction isn't symmetric
Unlike the product rule (which adds two terms, so order doesn't matter), the quotient rule subtracts — getting the order backward (f·g' − g·f' instead of g·f' − f·g') gives you the exact opposite sign, a completely wrong answer.
4
Always square the denominator
The denominator of your final answer is always g², regardless of how complex g itself is — don't forget this step even when the numerator work gets complicated.
Applied Walkthrough
1
Differentiate h(x) = x² / (x+1). Let f = x² (so f' = 2x) and g = x+1 (so g' = 1).
2
Apply "Low D-High minus High D-Low": g·f' − f·g' = (x+1)(2x) − (x²)(1).
3
Expand the numerator: 2x² + 2x − x² = x² + 2x.
4
Divide by g²: h'(x) = (x² + 2x) / (x+1)².
Exam Application
Exams test whether you can apply the quotient rule with the numerator terms in the correct order (since subtraction is order-sensitive), and whether you remember to square the denominator.
⚠ Common Trap
The most common trap is reversing the subtraction order in the numerator — writing f·g' − g·f' instead of g·f' − f·g'. Since subtraction isn't symmetric, this gives you the negative of the correct answer.
✓ Quick Self-Check
1. What is the quotient rule formula?
(f/g)' = (g·f' − f·g') / g².
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2. What does the memory phrase "Low D-High minus High D-Low" refer to?
Denominator times derivative of numerator, minus numerator times derivative of denominator.
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3. Differentiate h(x) = x² / (x+1).
h'(x) = (x² + 2x) / (x+1)².
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4. Why does the order of subtraction matter in the quotient rule, unlike the product rule?
Because subtraction isn't symmetric — reversing the order gives the negative of the correct answer, unlike addition in the product rule.
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5. What is the denominator of the final quotient rule answer?
g² — the original denominator squared.
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