# 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.

Source: https://practicecalculus.com/rules/quotient-rule
Updated: 2026-11-03

$$ \frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)\,g(x) - f(x)\,g'(x)}{[g(x)]^2} $$

## 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 $f \cdot g^{-1}$

## Key takeaways

- The derivative of $\frac{f}{g}$ is $\frac{f'g - fg'}{g^2}$. The minus sign means order matters.
- Mnemonic: low d-high minus high d-low, over the square of what's below.
- If the denominator is a constant or a single power of $x$, rewrite and skip the quotient rule.

The **quotient rule** finds the derivative of one function divided by another. Like the [product rule](/rules/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

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x)\,g(x) - f(x)\,g'(x)}{[g(x)]^2}$$

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 $f = x^2$, $f' = 2x$, $g = x + 1$, $g' = 1$:

$$\frac{2x(x+1) - x^2(1)}{(x+1)^2} = \frac{x^2 + 2x}{(x+1)^2}$$

**Example 2: simplify the numerator.** For $\frac{3x - 1}{x^2 + 1}$:

$$\frac{3(x^2 + 1) - (3x - 1)(2x)}{(x^2 + 1)^2} = \frac{3x^2 + 3 - 6x^2 + 2x}{(x^2 + 1)^2} = \frac{-3x^2 + 2x + 3}{(x^2 + 1)^2}$$

**Example 3: the derivative of $\tan x$.** Write $\tan x = \frac{\sin x}{\cos x}$:

$$\frac{\cos x \cdot \cos x - \sin x \cdot (-\sin x)}{\cos^2 x} = \frac{\cos^2 x + \sin^2 x}{\cos^2 x} = \frac{1}{\cos^2 x} = \sec^2 x$$

**Example 4: an exponential over a power.** For $\frac{e^x}{x}$:

$$\frac{e^x \cdot x - e^x \cdot 1}{x^2} = \frac{e^x(x - 1)}{x^2}$$

```graph
title: 'Graph of y = eˣ/x and its derivative eˣ(x − 1)/x²'
caption: 'The derivative is zero at $x = 1$, where $\frac{e^x}{x}$ has its minimum value $e$ on $x > 0$.'
x: [0.2, 3]
y: [-3, 8]
functions:
  - expr: e^x/x
    label: y = eˣ/x
  - expr: e^x*(x-1)/x^2
    label: derivative
    dashed: true
points:
  - [1, 2.718, '(1, e)']
```

## 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](/rules/constant-multiple-rule); the others use the [power rule](/rules/power-rule) and [sum rule](/rules/sum-and-difference-rule).

## Where the quotient rule comes from

Write $\frac{f}{g} = f \cdot g^{-1}$. The product rule and the [chain rule](/rules/chain-rule) give

$$f' \cdot g^{-1} + f \cdot (-g^{-2}g') = \frac{f'}{g} - \frac{fg'}{g^2} = \frac{f'g - fg'}{g^2}$$

So you can always fall back on the product rule if you forget the formula.

## Common mistakes

1. **Swapping the order in the numerator.** $fg' - f'g$ gives the negative of the right answer.
2. **Forgetting to square the denominator.** The bottom is $g^2$, not $g$.
3. **Dividing the derivatives.** $\left(\frac{f}{g}\right)' \ne \frac{f'}{g'}$.
4. **Expanding the denominator.** Leave $(x+1)^2$ factored. It keeps the answer simpler and makes zeros and sign changes easy to see.
5. **Confusing it with L'Hôpital's rule.** [L'Hôpital's rule](/rules/lhopitals-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](/blog/product-and-quotient-rules).

## Frequently asked questions

### What is the quotient rule?

The quotient rule says $\frac{d}{dx}\frac{f}{g} = \frac{f'g - fg'}{g^2}$. It gives the derivative of one function divided by another.

### What is the quotient rule mnemonic?

"Low d-high minus high d-low, over low squared." Low is the denominator $g$, high is the numerator $f$, and d means derivative of.

### Does order matter in the quotient rule?

Yes. Swapping the two terms in the numerator flips the sign of the answer. Always start with the derivative of the numerator times the denominator.

### When should you not use the quotient rule?

When the denominator is a constant, use the constant multiple rule. When it is a single power such as $x^3$, rewrite with a negative exponent and use the power rule.