词条 | Quotient rule |
释义 |
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions.[1][2][3] Let where both and are differentiable and The quotient rule states that the derivative of is Examples
ProofsProof from derivative definition and limit propertiesLet Applying the definition of the derivative and properties of limits gives the following proof. Proof using implicit differentiationLet so The product rule then gives Solving for and substituting back for gives: Proof using the chain ruleLet Then the product rule gives To evaluate the derivative in the second term, apply the power rule along with the chain rule: Finally, rewrite as fractions and combine terms to get Higher order formulasImplicit differentiation can be used to compute the {{mvar|n}}th derivative of a quotient (partially in terms of its first {{math|n − 1}} derivatives). For example, differentiating twice (resulting in ) and then solving for yields References1. ^{{cite book | last=Stewart | first=James | authorlink=James Stewart (mathematician) | title=Calculus: Early Transcendentals |publisher=Brooks/Cole | edition=6th | year=2008 | isbn=0-495-01166-5}} {{DEFAULTSORT:Quotient Rule}}2. ^{{cite book | last1=Larson | first1=Ron | authorlink=Ron Larson (mathematician)| last2=Edwards | first2=Bruce H. | title=Calculus | publisher=Brooks/Cole | edition=9th | year=2009 | isbn=0-547-16702-4}} 3. ^{{cite book | last1 = Thomas | first1 = George B. | last2=Weir | first2= Maurice D. | last3=Hass | first3=Joel | author3-link = Joel Hass | authorlink=George B. Thomas | title=Thomas' Calculus: Early Transcendentals | publisher=Addison-Wesley | year=2010 | edition=12th | isbn=0-321-58876-2}} 2 : Differentiation rules|Articles containing proofs |
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