2.9 The Quotient Rule

The Quotient Rule Formula

For f(x) = u(x)/v(x), the derivative is f'(x) = [u'v - uv'] / v². Remember it as "low-dee-high minus high-dee-low, over low squared." The order of subtraction in the numerator matters — swapping it gives the wrong sign!

When to Apply It

Use the quotient rule whenever a function is written as one expression divided by another, and it can't easily be rewritten as a product. Always identify u (numerator) and v (denominator) first, find u' and v' separately, then plug into the formula carefully.

Common Pitfalls

Students often forget to square the denominator or mix up the subtraction order. Also, simplify before differentiating if possible (e.g., splitting a fraction) — sometimes avoiding the quotient rule entirely makes the problem easier and less error-prone.

Key terms

Quotient Rule formula

If f(x) = u(x)/v(x), then f'(x) = [u'v - uv'] / v².

Mnemonic for Quotient Rule

"Low-dee-high minus high-dee-low, over low squared" (v·u' - u·v', all over v²).

Common quotient rule mistake

Forgetting to square the denominator, or swapping the order of subtraction in the numerator.

Derivative of tan(x) using quotient rule

tan(x) = sin(x)/cos(x); using quotient rule gives derivative = sec²(x).

When to consider avoiding the quotient rule

When the expression can be simplified or rewritten as a product (using negative exponents), the product rule may be simpler.

Try an AP-style question

d/dx [x/(x + 1)] = ?

1. 1/(x + 1)²
2. x/(x + 1)²
3. −1/(x + 1)²
4. 1