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Topics in Calculus
Fundamental theorem | Function | Limits of functions | Continuity | Calculus with polynomials
Differentiation
Product rule | Quotient rule | Chain rule | Implicit differentiation | Taylor's theorem
Integration
Integration by substitution | Integration by parts | Integration by trigonometric substitution | Solids of revolution | Integration by disks | Integration by cylindrical shells | Lists of integrals
Vector Calculus
Vector | Vector field | Matrix | Partial Derivative | Gradient | Flux | Divergence | Divergence Theorem | Del | Curl | Green's Theorem | Stokes' Theorem
Tensor Calculus
Tensor | Tensor field | Tensor product | Exterior power | Exterior Derivative | Covariant derivative | Manifold
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:

1 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:



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Quotient Rule Derivative X''2 4''x Denominator

Topics: Quotient Rule Derivative X''2 4''x Denominator H''(''x 1)2...

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Quotient space (linear algebra)  


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