Key takeaway: The quotient rule gives the derivative of one function divided by another: the bottom times the top’s derivative, minus the top times the bottom’s derivative, all over the bottom squared.
Grade 12 – AP Calculus ABAP Calculus AB Unit 2Calculus
Definition of Quotient Rule
The quotient rule differentiates a fraction whose numerator and denominator are both functions of \(x\). Because the denominator changes too, the derivative cannot be found by differentiating the top and bottom separately.
The formula comes from writing the quotient as a product, \(f\cdot g^{-1}\), and applying the product and chain rules. The result has a subtraction on top, so the order of the two terms matters.
Formula and notation
If \(y=\frac{f(x)}{g(x)}\) with \(g(x)\neq 0\), then
\[ y^{\prime}=\frac{f^{\prime}(x)\,g(x)-f(x)\,g^{\prime}(x)}{[g(x)]^2}. \]
Remember it as: low times d-high, minus high times d-low, over low squared.
Key facts
- The numerator is a difference, so swapping the two terms flips the sign of the answer.
- The denominator of the result is the original denominator squared, not its derivative.
- Simplify the numerator, but leave the squared denominator factored; it is easier to read.
- The quotient rule is also the standard way to differentiate any rational function.
Quotient Rule example
Find \(y^{\prime}\) for \(y=\frac{3x-1}{x^2+4}\), then evaluate it at \(x=2\).
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Take \(f=3x-1\), \(g=x^2+4\), so \(f^{\prime}=3\) and \(g^{\prime}=2x\).
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Apply the rule: \(y^{\prime}=\frac{3(x^2+4)-(3x-1)(2x)}{(x^2+4)^2}\).
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Simplify the numerator: \(3x^2+12-6x^2+2x=-3x^2+2x+12\).
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At \(x=2\): numerator \(-12+4+12=4\), denominator \(8^2=64\).
\(y^{\prime}=\frac{-3x^2+2x+12}{(x^2+4)^2}\), and at \(x=2\) the slope is \(\frac{4}{64}=\)\(\frac{1}{16}\).
Common mistake
Watch out
A student differentiates the same quotient as \(\frac{3}{2x}\), taking the derivative of the top over the derivative of the bottom. At \(x=2\) that gives \(\frac{3}{4}\).
Why it is wrong
Differentiating top and bottom separately is not a valid rule, as the value \(\frac{3}{4}\) shows against the correct \(\frac{1}{16}\). Use the full quotient rule, with both cross products and the squared denominator.
Related terms and lessons
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