Quotient rule is one of 9 derivatives formulas in the calculus section of this library, and it is used at ap · university level.
Why quotient rule works
A quotient is really the top function multiplied by one over the bottom one. Growing the bottom shrinks the fraction, which is why the g-prime term arrives with a subtraction, and putting both pieces over a single denominator is what produces the square underneath.
What each symbol means
$f$ is numerator and $g$ denominator.
Quotient rule: when it holds
$f,g$ must be differentiable and $g(x)\ne0$.
When it stops applying
Where the bottom function is 0 the fraction itself has no value, so there is nothing there to differentiate and the rule returns a division by 0. An expression like (x²−1)/(x−1) has to be cancelled down to x+1 before you can talk about its slope at x = 1.
Quotient rule: a worked example
$\frac{d}{dx}(x/e^x)=(e^x-xe^x)/e^{2x}=(1-x)e^{-x}$.
The mistake to avoid
What people do: Starting the numerator with the top times the derivative of the bottom, so the two terms are swapped.
Why it goes wrong: The whole numerator then comes out as the exact negative of the truth, and every sign in the answer is backwards.
Do this instead: Start by differentiating the top: bottom times derivative of top, minus top times derivative of bottom, all over the bottom squared.
Quotient rule: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. $f$ is numerator and $g$ denominator.
- Check the conditions before substituting. $f,g$ must be differentiable and $g(x)\ne0$.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most calculus slips.
Where this formula fits
- Subject
- Calculus formulas — 42 entries in this library
- Topic
- Derivatives
- Level
- AP · University
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where quotient rule comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Derivative Rules — the lesson behind this formula: differentiate sums, products, quotients, and compositions.
- Calculus Calculator — check your substitution and the value it produces.
- Study calculus — the subject guide that explains the ideas these formulas compress.
- Calculus Practice — questions that make you retrieve the formula instead of recognising it.
- All 42 calculus formulas — the full grouped reference, or the complete formula library.
Questions about quotient rule
Can I dodge this rule by rewriting with a negative exponent?
Often yes. Writing x over e to the x as x times e to the −x lets the product rule do the job, and agreeing answers from both routes is a good check.
Why does the bottom get squared?
Differentiating one over g already produces g squared underneath, and collecting both terms over one common denominator forces every piece to sit over that same square.
Does the squared denominator ever need absolute value bars?
No. Squaring has already made that factor positive, so whether the final answer is positive or negative is decided entirely by the numerator.
What happens when the top is just a number?
Its derivative is 0, the first term vanishes, and the rule shrinks to minus the constant times the derivative of the bottom over the bottom squared. That is the fast way to handle 5 over x cubed.