Quotient Rule

Quotient Rule

A formula for the derivative of the quotient of two functions.

Quotient Rule formula (u/v)'=u'v-uv'/v², with examples for (1-5x)/(4x+7) and d/dx(sin x/cos x)=sec²x

See also

Key Formula

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

Where:

$f(x)$ = The numerator function (the function on top)
$g(x)$ = The denominator function (the function on the bottom)
$f'(x)$ = The derivative of the numerator function
$g'(x)$ = The derivative of the denominator function

Worked Example

Problem: Find the derivative of h(x) = x² / (3x + 1).

Step 1: Identify the numerator and denominator functions.

f(x) = x^2, \quad g(x) = 3x + 1

Step 2: Find the derivatives of each function separately.

f'(x) = 2x, \quad g'(x) = 3

Step 3: Substitute into the Quotient Rule formula: denominator times derivative of numerator, minus numerator times derivative of denominator, all over the denominator squared.

h'(x) = \frac{(3x+1)(2x) - (x^2)(3)}{(3x+1)^2}

Step 4: Expand the numerator.

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

Step 5: Simplify by combining like terms in the numerator.

h'(x) = \frac{3x^2 + 2x}{(3x+1)^2}

Answer:

h'(x) = \frac{3x^2 + 2x}{(3x+1)^2}

Another Example

This example involves a trigonometric function divided by a simple polynomial, showing how the Quotient Rule works with non-polynomial functions. It also highlights that the result cannot always be simplified into a cleaner form.

Problem: Find the derivative of y = sin(x) / x.

Step 1: Identify the numerator and denominator functions.

f(x) = \sin(x), \quad g(x) = x

Step 2: Differentiate each function.

f'(x) = \cos(x), \quad g'(x) = 1

Step 3: Apply the Quotient Rule.

y' = \frac{x \cdot \cos(x) - \sin(x) \cdot 1}{x^2}

Step 4: Write the simplified result.

y' = \frac{x\cos(x) - \sin(x)}{x^2}

Answer:

y' = \frac{x\cos(x) - \sin(x)}{x^2}

Frequently Asked Questions

What is an easy way to remember the Quotient Rule?

A popular mnemonic is "low d-high minus high d-low, over the square of what's below." Here, "low" is the denominator g(x), "high" is the numerator f(x), and "d-high" and "d-low" mean their derivatives. This phrase matches the formula exactly: g·f' − f·g', all over g².

When should you use the Quotient Rule instead of the Product Rule?

Use the Quotient Rule when a function is written as a fraction with one function divided by another. You could rewrite f(x)/g(x) as f(x)·[g(x)]⁻¹ and use the Product Rule combined with the Chain Rule instead, but the Quotient Rule is often more direct. If the denominator is just a constant, you can simply factor it out and avoid the Quotient Rule entirely.

Does the order of subtraction matter in the Quotient Rule?

Yes, the order is critical. The formula is g·f' minus f·g', not the other way around. Reversing the subtraction changes the sign of the derivative and gives you a wrong answer. This is the single most common mistake students make with the Quotient Rule.

Quotient Rule vs. Product Rule

	Quotient Rule	Product Rule
Definition	Finds the derivative of f(x)/g(x)	Finds the derivative of f(x)·g(x)
Formula	[g·f' − f·g'] / g²	f·g' + g·f'
Subtraction involved?	Yes — order of terms matters	No — uses addition, so order doesn't matter
Denominator in result	Always has [g(x)]² in the denominator	No denominator is introduced
When to use	When one function is divided by another	When two functions are multiplied together

Why It Matters

The Quotient Rule appears constantly in AP Calculus and university-level calculus courses whenever you need to differentiate rational expressions or ratios of functions. It is essential for finding slopes of tangent lines, rates of change, and critical points of functions expressed as fractions. Many real-world models — such as concentration of a drug over time or efficiency ratios in physics — naturally involve quotients, making this rule a practical tool beyond the classroom.

Common Mistakes

Mistake: Reversing the subtraction order in the numerator, writing f·g' − g·f' instead of g·f' − f·g'.

Correction: Always start with the denominator times the derivative of the numerator first. The mnemonic 'low d-high minus high d-low' keeps the order straight: the denominator (low) comes first.

Mistake: Forgetting to square the denominator, writing g(x) instead of [g(x)]² on the bottom.

Correction: The Quotient Rule always divides by the square of the original denominator. After applying the formula, double-check that your final answer has [g(x)]² in the denominator.

Related Terms

Derivative — The Quotient Rule computes a derivative
Derivative Rules — The Quotient Rule is one of several rules
Product Rule — Companion rule for products of functions
Chain Rule — Often used together with the Quotient Rule
Quotient — The type of expression this rule applies to
Function — The numerator and denominator are both functions
Formula — The Quotient Rule is a derivative formula
Power Rule — Basic rule often used within Quotient Rule steps