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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
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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:
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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:
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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:
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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:
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Proof
A proof of this rule can be derived from Newton's difference quotient:
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- where h(x) ≠ 0 and g and h are differentiable.
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Alternate Informal Proof
Using only the product rule:
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- g(x) = f(x)h(x)
- g'(x) = f'(x)h(x) +
f(x)h'(x)
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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