HOME

TheInfoList



OR:

In mathematics, specifically in calculus and
complex analysis Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebrai ...
, the logarithmic derivative of a function ''f'' is defined by the formula \frac where f' is the
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of ''f''. Intuitively, this is the infinitesimal
relative change In any quantitative science, the terms relative change and relative difference are used to compare two quantities while taking into account the "sizes" of the things being compared, i.e. dividing by a ''standard'' or ''reference'' or ''starting'' va ...
in ''f''; that is, the infinitesimal absolute change in ''f,'' namely f', scaled by the current value of ''f.'' When ''f'' is a function ''f''(''x'') of a real variable ''x'', and takes
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
, strictly
positive Positive is a property of positivity and may refer to: Mathematics and science * Positive formula, a logical formula not containing negation * Positive number, a number that is greater than 0 * Plus sign, the sign "+" used to indicate a posi ...
values, this is equal to the derivative of ln(''f''), or the
natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...
of ''f''. This follows directly from the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
: \frac\ln f(x) = \frac \frac


Basic properties

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 (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . 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 \frac = \frac = \frac + \frac . 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). A
corollary In mathematics and logic, a corollary ( , ) is a theorem of less importance which can be readily deduced from a previous, more notable statement. A corollary could, for instance, be a proposition which is incidentally proved while proving anothe ...
to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function: \frac = \frac = -\frac , just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number. More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor: \frac = \frac = \frac - \frac , just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor. 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: \frac = \frac = k \frac , just as the logarithm of a power is the product of the exponent and the logarithm of the base. In summary, both derivatives and logarithms have a
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
, a
reciprocal rule In calculus, the reciprocal rule gives the derivative of the reciprocal of a function ''f'' in terms of the derivative of ''f''. The reciprocal rule can be used to show that the power rule holds for negative exponents if it has already been ...
, a
quotient rule In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriva ...
, and a
power rule In calculus, the power rule is used to differentiate functions of the form f(x) = x^r, whenever r is a real number. Since differentiation is a linear operation on the space of differentiable functions, polynomials can also be differentiated u ...
(compare the list of logarithmic identities); each pair of rules is related through the logarithmic derivative.


Computing ordinary derivatives using logarithmic derivatives

Logarithmic derivatives can simplify the computation of derivatives requiring the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
while producing the same result. The procedure is as follows: Suppose that and that we wish to compute . Instead of computing it directly as , we compute its logarithmic derivative. That is, we compute: \frac = \frac + \frac. Multiplying through by ƒ computes : f' = f\cdot\left(\frac + \frac\right). This technique is most useful when ƒ is a product of a large number of factors. This technique makes it possible to compute by computing the logarithmic derivative of each factor, summing, and multiplying by . For example, we can compute the logarithmic derivative of e^\frac to be 2x + \frac + \frac - \frac.


Integrating factors

The logarithmic derivative idea is closely connected to the
integrating factor In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calculu ...
method for
first-order differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
s. In operator terms, write D = \frac and let ''M'' denote the operator of multiplication by some given function ''G''(''x''). Then M^ D M can be written (by the
product rule In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions. For two functions, it may be stated in Lagrange's notation as (u \cdot v)' = u ' \cdot v + ...
) as D + M^ where M^ now denotes the multiplication operator by the logarithmic derivative \frac In practice we are given an operator such as D + F = L and wish to solve equations L(h) = f for the function ''h'', given ''f''. This then reduces to solving \frac = F which has as solution \exp \textstyle ( \int F ) with any
indefinite integral In calculus, an antiderivative, inverse derivative, primitive function, primitive integral or indefinite integral of a function is a differentiable function whose derivative is equal to the original function . This can be stated symbolically ...
of ''F''.


Complex analysis

The formula as given can be applied more widely; for example if ''f''(''z'') is a
meromorphic function In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the complex plane is a function that is holomorphic on all of ''D'' ''except'' for a set of isolated points, which are poles of the function. The te ...
, it makes sense at all complex values of ''z'' at which ''f'' has neither a zero nor a 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 :''zn'' with ''n'' an integer, . The logarithmic derivative is then n/z 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 Residue may refer to: Chemistry and biology * An amino acid, within a peptide chain * Crop residue, materials left after agricultural processes * Pesticide residue, refers to the pesticides that may remain on or in food after they are applie ...
''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 In the mathematical field of complex analysis, contour integration is a method of evaluating certain integrals along paths in the complex plane. Contour integration is closely related to the calculus of residues, a method of complex analysis. ...
. In the field of
Nevanlinna Theory In the mathematical field of complex analysis, Nevanlinna theory is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl called it "one of the few great mathematical events of (the twentieth) century ...
, 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 m(r,h'/h) = S(r,h) = o(T(r,h)).


The multiplicative group

Behind the use of the logarithmic derivative lie two basic facts about ''GL''1, that is, the multiplicative group of
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one- dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Ever ...
s or other
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
. The
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and return ...
X\frac is
invariant Invariant and invariance may refer to: Computer science * Invariant (computer science), an expression whose value doesn't change during program execution ** Loop invariant, a property of a program loop that is true before (and after) each iterat ...
under dilation (replacing ''X'' by ''aX'' for ''a'' constant). And the
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
\frac is likewise invariant. For functions ''F'' into GL1, the formula \frac is therefore a ''
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
'' of the invariant form.


Examples

* Exponential growth and
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
are processes with constant logarithmic derivative. * In
mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
, the
Greek Greek may refer to: Greece Anything of, from, or related to Greece, a country in Southern Europe: *Greeks, an ethnic group. *Greek language, a branch of the Indo-European language family. **Proto-Greek language, the assumed last common ancestor ...
 ''λ'' is the logarithmic derivative of derivative price with respect to underlying price. * In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...
, the
condition number In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the inpu ...
is the infinitesimal relative change in the output for a relative change in the input, and is thus a ratio of logarithmic derivatives.


See also

* *


References

{{Calculus topics Differential calculus Complex analysis