Quotient Rule for Derivatives: Formula, Mnemonic and Examples
The quotient rule: (f/g)' = (f'g − fg')/g². The formula, the low d-high mnemonic, worked examples, when to avoid it and common mistakes.
Derivative rules Updated 2 min read
At a glance
- Applies to
- A quotient of two differentiable functions, wherever g(x) \ne 0
- In words
- Low d-high minus high d-low, over low squared
- Order matters
- Yes, the numerator has a subtraction, so start with f'g
- Avoid it when
- The denominator is a constant or a single power of x
- Derived from
- The product rule and chain rule applied to f \cdot g^{-1}
On this page
The quotient rule finds the derivative of one function divided by another. Like the product rule, it is not what you might first guess: the derivative of \frac{f}{g} is not \frac{f'}{g'}. The numerator has a subtraction, so the order of the terms matters.
The quotient rule formula
If f and g are differentiable and g(x) \ne 0, then
The classic mnemonic, with "high" as the numerator and "low" as the denominator:
Low d-high minus high d-low, over low squared.
Worked examples
Example 1: a rational function. For \frac{x^2}{x + 1}, take f = x^2, f' = 2x, g = x + 1, g' = 1:
Example 2: simplify the numerator. For \frac{3x - 1}{x^2 + 1}:
Example 3: the derivative of \tan x. Write \tan x = \frac{\sin x}{\cos x}:
Example 4: an exponential over a power. For \frac{e^x}{x}:
When to skip the quotient rule
A fraction does not always need the quotient rule. Rewriting is often faster and less error-prone:
| Function | Rewrite | Derivative |
|---|---|---|
| \frac{x^3}{5} | \frac{1}{5}x^3 | \frac{3}{5}x^2 |
| \frac{4}{x^2} | 4x^{-2} | -\frac{8}{x^3} |
| \frac{x^3 + x}{x} | x^2 + 1 (for x \ne 0) | 2x |
| \frac{2}{\sqrt{x}} | 2x^{-1/2} | -x^{-3/2} |
The first uses the constant multiple rule; the others use the power rule and sum rule.
Where the quotient rule comes from
Write \frac{f}{g} = f \cdot g^{-1}. The product rule and the chain rule give
So you can always fall back on the product rule if you forget the formula.
Common mistakes
- Swapping the order in the numerator. fg' - f'g gives the negative of the right answer.
- Forgetting to square the denominator. The bottom is g^2, not g.
- Dividing the derivatives. \left(\frac{f}{g}\right)' \ne \frac{f'}{g'}.
- Expanding the denominator. Leave (x+1)^2 factored. It keeps the answer simpler and makes zeros and sign changes easy to see.
- Confusing it with L'Hôpital's rule. L'Hôpital's rule differentiates the top and bottom separately to find a limit. The quotient rule finds a derivative.
For a longer walkthrough of both rules together, read the full guide to the product and quotient rules.
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The quotient rule
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