Quotient Rule (calculus)
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:
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.
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.
Referenced By
Calculus | Derivative | Derivative (calculus) | Differentation | Differentiable function | Differential calculus | Leibniz' law | Product Rule (Calculus) | Product rule | Taylor's theorem | Taylors theorem
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