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| {{Unreferenced|date=December 2009}}
| | My name: Teresita Bohannon<br>Age: 20<br>Country: Germany<br>Home town: Woldegk <br>Post code: 17345<br>Street: Chausseestr. 64<br><br>Here is my website; ovarian cyst surgery ([http://www.math-labo.info/ www.math-labo.info]) |
| In [[mathematics]], specifically in [[calculus]] and [[complex analysis]], the '''logarithmic derivative''' of a [[function (mathematics)|function]] ''f'' is defined by the formula
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| : <math> \frac{f'}{f} \! </math>
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| where ''f'' ′ is the [[derivative]] of ''f''. Intuitively, this is the infinitesimal [[relative change]] in ''f'' – it is the infinitesimal absolute change in ''f,'' namely <math>f',</math> scaled by the current value of ''f.''
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| When ''f'' is a function ''f''(''x'') of a real variable ''x'', and takes [[real numbers|real]], strictly [[Positive number|positive]] values, this is equal to the derivative of ln(''f''); or, the derivative of the [[natural logarithm]] of ''f''. This follows directly from the [[chain rule]].
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| ==Basic properties==
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| Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does ''not'' take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have
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| : <math> (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' .\! </math> | |
| So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the [[Leibniz law]] for the derivative of a product to get
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| : <math> \frac{(uv)'}{uv} = \frac{u'v + uv'}{uv} = \frac{u'}{u} + \frac{v'}{v} .\! </math> | |
| Thus, it is true for ''any'' function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).
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| A [[corollary]] to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:
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| : <math> \frac{(1/u)'}{1/u} = \frac{-u'/u^{2}}{1/u} = -\frac{u'}{u} ,\! </math>
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| just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.
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| More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor:
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| : <math> \frac{(u/v)'}{u/v} = \frac{(u'v - uv')/v^{2}}{u/v} = \frac{u'}{u} - \frac{v'}{v} ,\! </math> | |
| just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.
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| Generalising in another direction, the logarithmic derivative of a power (with constant real exponent) is the product of the exponent and the logarithmic derivative of the base:
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| : <math> \frac{(u^{k})'}{u^{k}} = \frac {ku^{k-1}u'}{u^{k}} = k \frac{u'}{u} ,\! </math> | |
| just as the logarithm of a power is the product of the exponent and the logarithm of the base.
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| In summary, both derivatives and logarithms have a [[product rule]], a [[reciprocal rule]], a [[quotient rule]], and a [[power rule]] (compare the [[list of logarithmic identities]]); each pair of rules is related through the logarithmic derivative.
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| ==Computing ordinary derivatives using logarithmic derivatives==
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| {{Main|Logarithmic differentiation}}
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| Logarithmic derivatives can simplify the computation of derivatives requiring the product rule. The procedure is as follows: Suppose that {{Nowrap|1=ƒ(''x'') = ''u''(''x'')''v''(''x'')}} and that we wish to compute {{Nowrap|ƒ'(''x'')}}. Instead of computing it directly, we compute its logarithmic derivative. That is, we compute:
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| :<math>\frac{f'}{f} = \frac{u'}{u} + \frac{v'}{v}.</math>
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| Multiplying through by ƒ computes {{Nowrap|ƒ'}}:
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| :<math>f' = f\left(\frac{u'}{u} + \frac{v'}{v}\right).</math>
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| This technique is most useful when ƒ is a product of a large number of factors. This technique makes it possible to compute {{Nowrap|ƒ'}} by computing the logarithmic derivative of each factor, summing, and multiplying by ƒ.
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| ==Integrating factors==
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| The logarithmic derivative idea is closely connected to the [[integrating factor]] method for [[first-order differential equation]]s. In [[Operator (mathematics)|operator]] terms, write
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| :''D'' = ''d''/''dx''
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| and let ''M'' denote the operator of multiplication by some given function ''G''(''x''). Then
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| :''M''<sup>−1</sup>''DM''
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| can be written (by the [[product rule]]) as
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| :''D'' + ''M*''
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| where ''M*'' now denotes the multiplication operator by the logarithmic derivative
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| :''G''′/''G''.
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| In practice we are given an operator such as
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| :''D'' + ''F'' = ''L''
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| and wish to solve equations
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| :''L''(''h'') = ''f''
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| for the function ''h'', given ''f''. This then reduces to solving
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| :''G''′/''G'' = ''F''
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| which has as solution
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| :exp(∫''F'')
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| with any [[indefinite integral]] of ''F''.
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| ==Complex analysis==
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| The formula as given can be applied more widely; for example if ''f''(''z'') is a [[meromorphic function]], it makes sense at all complex values of ''z'' at which ''f'' has neither a zero nor a [[Pole (complex analysis)|pole]]. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case
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| :''z<sup>n</sup>'' | |
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| with ''n'' an integer, ''n'' ≠ 0. The logarithmic derivative is then
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| :''n''/''z'';
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| and one can draw the general conclusion that for ''f'' meromorphic, the singularities of the logarithmic derivative of ''f'' are all ''simple'' poles, with [[residue (complex analysis)|residue]] ''n'' from a zero of order ''n'', residue −''n'' from a pole of order ''n''. See [[argument principle]]. This information is often exploited in [[contour integration]].
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| In the field of [[Nevanlinna Theory]], an important lemma states that the proximity function of a logarithmic derivative is small with respect to the Nevanlinna Characteristic of the original function, for instance <math>m(r,h'/h) = S(r,h) = o(T(r,h))</math>.
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| ==The multiplicative group==
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| Behind the use of the logarithmic derivative lie two basic facts about ''GL''<sub>1</sub>, that is, the multiplicative group of [[real number]]s or other [[field (mathematics)|field]]. The [[differential operator]]
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| : <math> X\frac{d}{dX} </math>
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| is [[Invariant (mathematics)|invariant]] under 'translation' (replacing ''X'' by ''aX'' for ''a'' constant). And the [[differential form]]
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| :''dX/X''
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| is likewise invariant. For functions ''F'' into ''GL''<sub>1</sub>, the formula
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| :''dF/F''
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| is therefore a ''[[pullback (differential geometry)|pullback]]'' of the invariant form.
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| ==Examples==
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| * [[Exponential growth]] and [[exponential decay]] are processes with constant logarithmic derivative.
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| * In [[mathematical finance]], the [[The Greeks|Greek]] ''λ'' is the logarithmic derivative of derivative price with respect to underlying price.
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| * In [[numerical analysis]], the [[condition number]] is the infinitesimal relative change in the output for a relative change in the input, and is thus a ratio of logarithmic derivatives.
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| {{DEFAULTSORT:Logarithmic Derivative}}
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| [[Category:Differential calculus]]
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| [[Category:Complex analysis]]
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My name: Teresita Bohannon
Age: 20
Country: Germany
Home town: Woldegk
Post code: 17345
Street: Chausseestr. 64
Here is my website; ovarian cyst surgery (www.math-labo.info)