词条 | Product rule |
释义 |
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated as or in Leibniz's notation In differentials notation, this can be written as In Leibniz's notation, the derivative of the product of three functions (not to be confused with Euler's triple product rule) is DiscoveryDiscovery of this rule is credited to Gottfried Leibniz, who demonstrated it using differentials.[1] (However, J. M. Child, a translator of Leibniz's papers,[2] argues that it is due to Isaac Barrow.) Here is Leibniz's argument: Let u(x) and v(x) be two differentiable functions of x. Then the differential of uv is Since the term du·dv is "negligible" (compared to du and dv), Leibniz concluded that and this is indeed the differential form of the product rule. If we divide through by the differential dx, we obtain which can also be written in Lagrange's notation as Examples
ProofsProof by factoring (Proof from first principles)Let {{math|h(x) {{=}} f(x)g(x)}} and suppose that {{mvar|f}} and {{mvar|g}} are each differentiable at {{mvar|x}}. We want to prove that {{mvar|h}} is differentiable at {{mvar|x}} and that its derivative, {{math|{{prime|h}}(x)}}, is given by {{math|{{prime|f}}(x)g(x) + f(x){{prime|g}}(x)}}. To do this, (which is zero, and thus does not change the value) is added to the numerator to permit its factoring, and then properties of limits are used. Brief proofBy definition, if are differentiable at then we can write such that , also written . Then: Taking the limit for small gives the result. Quarter squaresThere is a proof using quarter square multiplication which relies on the chain rule and on the properties of the quarter square function (shown here as q, i.e., with ): Differentiating both sides: Chain ruleThe product rule can be considered a special case of the chain rule for several variables. Non-standard analysisLet u and v be continuous functions in x, and let dx, du and dv be infinitesimals within the framework of non-standard analysis, specifically the hyperreal numbers. Using st to denote the standard part function that associates to a finite hyperreal number the real infinitely close to it, this gives This was essentially Leibniz's proof exploiting the transcendental law of homogeneity (in place of the standard part above). Smooth infinitesimal analysisIn the context of Lawvere's approach to infinitesimals, let dx be a nilsquare infinitesimal. Then du = u' dx and dv = v' dx, so that since Generalizations{{unreferenced section|date=July 2013}}A product of more than two factorsThe product rule can be generalized to products of more than two factors. For example, for three factors we have . For a collection of functions , we have Higher derivativesIt can also be generalized to the general Leibniz rule for the nth derivative of a product of two factors, by symbolically expanding according to the binomial theorem: Applied at a specific point x, the above formula gives: Furthermore, for the nth derivative of an arbitrary number of factors: Higher partial derivativesFor partial derivatives, we have where the index S runs through the whole list of 2n subsets of {1, ..., n}. For example, when n = 3, then Banach spaceSuppose X, Y, and Z are Banach spaces (which includes Euclidean space) and B : X × Y → Z is a continuous bilinear operator. Then B is differentiable, and its derivative at the point (x,y) in X × Y is the linear map D(x,y)B : X × Y → Z given by Derivations in abstract algebraIn abstract algebra, the product rule is used to define what is called a derivation, not vice versa. Vector functionsThe product rule extends to scalar multiplication, dot products, and cross products of vector functions. For scalar multiplication: For dot products: For cross products: Note: cross products are not commutative, i.e. , instead products are anticommutative, so it can be written as Scalar fieldsFor scalar fields the concept of gradient is the analog of the derivative: ApplicationsAmong the applications of the product rule is a proof that when n is a positive integer (this rule is true even if n is not positive or is not an integer, but the proof of that must rely on other methods). The proof is by mathematical induction on the exponent n. If n = 0 then xn is constant and nxn − 1 = 0. The rule holds in that case because the derivative of a constant function is 0. If the rule holds for any particular exponent n, then for the next value, n + 1, we have Therefore if the proposition is true for n, it is true also for n + 1, and therefore for all natural n. References1. ^{{cite journal |author=Michelle Cirillo|journal=The Mathematics Teacher |volume=101 |issue=1 |pages=23–27 |date=August 2007 |title=Humanizing Calculus|url=http://www.nctm.org/publications/article.aspx?id=19302|subscription=yes}} 2. ^{{citation|first=G. W.|last=Leibniz|url=http://dynref.engr.illinois.edu/rvc_Child_1920.pdf|title= The Early Mathematical Manuscripts of Leibniz|translator=J.M. Child|year=2005|origyear=1920|publisher=Dover|page= 28, footnote 58|isbn=978-0-486-44596-0}} External links
2 : Differentiation rules|Articles containing proofs |
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