Redirected from Quotient Rule (calculus)
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:
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):
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).
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.
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