词条 | Logarithmic differentiation |
释义 |
In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f,[1] The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. This usually occurs in cases where the function of interest is composed of a product of a number of parts, so that a logarithmic transformation will turn it into a sum of separate parts (which is much easier to differentiate). It can also be useful when applied to functions raised to the power of variables or functions. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions.[2][3] The principle can be implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero. OverviewFor a function logarithmic differentiation typically begins by taking the natural logarithm, or the logarithm to the base e, on both sides, remembering to take absolute values:[4] After implicit differentiation:[5] Multiplication by y is then done to eliminate 1/y and leave only dy/dx on the left-hand side: The method is used because the properties of logarithms provide avenues to quickly simplify complicated functions to be differentiated.[6] These properties can be manipulated after the taking of natural logarithms on both sides and before the preliminary differentiation. The most commonly used logarithm laws are[3] General caseUsing capital pi notation, Application of natural logarithms results in (with capital sigma notation) and after differentiation, Rearrange to get the derivative of the original function, Higher order derivativesUsing Faà di Bruno's formula, the n-th order logarithmic derivative is, Using this, the first four derivatives are, ApplicationsProductsA natural logarithm is applied to a product of two functions to transform the product into a sum Differentiating by applying the chain and the sum rules yields and, after rearranging, yields[7] QuotientsA natural logarithm is applied to a quotient of two functions to transform the division into a subtraction Differentiating by applying the chain and the sum rules yields and, after rearranging, yields After multiplying out and using the common denominator formula the result is the same as after applying the quotient rule directly to . Composite exponentFor a function of the form The natural logarithm transforms the exponentiation into a product Differentiating by applying the chain and the product rules yields and, after rearranging, yields The same result can be obtained by rewriting f in terms of exp and applying the chain rule. See also
Notes1. ^{{cite book|title=Calculus demystified|pages=170|first=Steven G.|last=Krantz|publisher=McGraw-Hill Professional|year=2003|isbn=0-07-139308-0}} 2. ^{{cite book|title=Golden Differential Calculus|pages=282|author=N.P. Bali|publisher=Firewall Media|year=2005|isbn=81-7008-152-1}} 3. ^1 {{cite book|title=Higher Engineering Mathematics|first=John|last=Bird|pages=324|publisher=Newnes|year=2006|isbn=0-7506-8152-7}} 4. ^{{cite book|title=Schaum's Outline of Theory and Problems of Calculus for Business, Economics, and the Social Sciences|first=Edward T.|last=Dowling|publisher=McGraw-Hill Professional|year=1990|isbn=0-07-017673-6|pages=160}} 5. ^{{cite book|title=Calculus of One Variable|first=Keith|last=Hirst|pages=97|publisher=Birkhäuser|year=2006|isbn=1-85233-940-3}} 6. ^{{cite book|title=Calculus, single variable|first=Brian E.|last=Blank|pages=457|publisher=Springer|year=2006|isbn=1-931914-59-1}} 7. ^{{cite book|title=An Elementary Treatise on the Differential Calculus|first=Benjamin|last=Williamson|publisher=BiblioBazaar, LLC|year=2008|pages=25–26|isbn=0-559-47577-2}} 1 : Differential calculus |
随便看 |
|
开放百科全书收录14589846条英语、德语、日语等多语种百科知识,基本涵盖了大多数领域的百科知识,是一部内容自由、开放的电子版国际百科全书。