Quotient rule

Quotient rule

In calculus, the quotient rule is a method of finding the derivative of a function which is the quotient of two other functions for which derivatives exist.

If the function one wishes to differentiate, f(x), can be written as

and h(x) ≠ 0; then, the rule states that the derivative of g(x) / h(x) is equal to the denominator times the derivative of the numerator, minus the numerator times the derivative of the denominator, all divided by the square of the denominator:

Or more precisely; for all x'\' in some open set containing the number a, with h(a) ≠ 0; and, such that g '(a) and h '(a) both exist; then, f '(a'') exists as well:

Table of contents

1 Examples
2 Informal Proof

2.1 Alternate Informal Proof

Examples

The derivative of (4x - 2) / (x² + 1) = [(x² + 1)(4) - (4x - 2)(2x)] / (x² + 1)² = [(4x² + 4) - (8x² - 4x)] / (x² + 1)² = [-4x² + 4x + 4] / (x² + 1)²

The derivative of [sin(x)] / x² (when x ≠ 0) is ([cos(x)]x² - [sin(x)](2x)) / x⁴.

For more information regarding the derivatives of trigonometric functions, see: derivative.

Another example is:

whereas g(x) = 2x² and h(x) = x³, and g′(x) = 4x and h′(x) = 3x².

The derivative of f(x) is determined as follows:

Informal Proof

A proof of this rule can be derived from Newton's difference quotient: The derivative of [f(x)] / [g(x)] = (the limit as h approaches 0):

having multiplied the fraction by: g(x)(x + Δx) / g(x)(x + Δx)

adding and subtracting the same value to allow for an algebraic manipulation

seperating into groups with common multiples

which for Δx → 0 converges to

To turn this into a proper proof, one has to pick Δx so small that the denominators are all non-zero (and one has to argue that this is always possible because the involved functions are continuous).

Alternate Informal Proof

Using only the product rule:

The rest is simple algebra to make f'(x) the only term on the left hand side of the equation and to remove f(x) from the right side of the equation.