Logarithmic derivative
Encyclopedia
In mathematics
, specifically in calculus
and complex analysis
, the logarithmic derivative of a function
f is defined by the formula
where f ′ is the derivative
of f.
When f is a function f(x) of a real variable x, and takes real, strictly 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
.
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
Thus, it's 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).
Similarly (in fact this is a consequence), the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:
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:
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:
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
, 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.
Multiplying through by ƒ computes :
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 ƒ.
method for first-order differential equations. In operator terms, write
and let M denote the operator of multiplication by some given function G(x). Then
can be written (by the product rule
) as
where M* now denotes the multiplication operator by the logarithmic derivative
In practice we are given an operator such as
and wish to solve equations
for the function h, given f. This then reduces to solving
which has as solution
with any indefinite integral of F.
, 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
with n an integer, n ≠ 0. The logarithmic derivative is then
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
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.
s or other field
. The differential operator
is invariant
under 'translation' (replacing X by aX for a constant). And the differential form
is likewise invariant. For functions F into GL1, the formula
is therefore a pullback of the invariant form.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, specifically in calculus
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
and complex analysis
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 useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, the logarithmic derivative of a function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
f is defined by the formula
where f ′ is the derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
of f.
When f is a function f(x) of a real variable x, and takes real, strictly positive values, this is equal to the derivative of ln(f); or, the derivative of the natural logarithm
Natural logarithm
The natural logarithm is the logarithm to the base e, where e is an irrational and transcendental constant approximately equal to 2.718281828...
of f. This follows directly from the chain rule
Chain rule
In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...
.
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
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
Thus, it's 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).
Similarly (in fact this is a consequence), the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:
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:
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:
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
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
, a reciprocal rule
Reciprocal rule
In calculus, the reciprocal rule is a shorthand method of finding the derivative of a function that is the reciprocal of a differentiable function, without using the quotient rule or chain rule....
, a quotient rule, and a power rule (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. The procedure is as follows: Suppose that and that we wish to compute . Instead of computing it directly, we compute its logarithmic derivative. That is, we compute:Multiplying through by ƒ computes :
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 ƒ.
Integrating factors
The logarithmic derivative idea is closely connected to the integrating factorIntegrating 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 calculus, in this case often multiplying through by an...
method for first-order differential equations. In operator terms, write
- D = d/dx
and let M denote the operator of multiplication by some given function G(x). Then
- M−1DM
can be written (by the product rule
Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions. It may be stated thus:'=f'\cdot g+f\cdot g' \,\! or in the Leibniz notation thus:...
) as
- D + M*
where M* now denotes the multiplication operator by the logarithmic derivative
- G′/G.
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
- G′/G = F
which has as solution
- exp(∫F)
with any indefinite integral of F.
Complex analysis
The formula as given can be applied more widely; for example if f(z) is a meromorphic functionMeromorphic function
In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...
, 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, n ≠ 0. 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 (complex analysis)
In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities...
n from a zero of order n, residue −n from a pole of order n. See argument principle
Argument principle
In complex analysis, the argument principle determines the difference between the number of zeros and poles of a meromorphic function by computing a contour integral of the function's logarithmic derivative....
. This information is often exploited in contour integration.
The multiplicative group
Behind the use of the logarithmic derivative lie two basic facts about GL1, that is, the multiplicative group of real numberReal number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s or other field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
. The differential operator
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, accepting a function and returning another .This article considers only linear operators,...
is invariant
Invariant (mathematics)
In mathematics, an invariant is a property of a class of mathematical objects that remains unchanged when transformations of a certain type are applied to the objects. The particular class of objects and type of transformations are usually indicated by the context in which the term is used...
under 'translation' (replacing X by aX for a constant). And the differential form
Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a better definition for integrands in calculus...
- dX/X
is likewise invariant. For functions F into GL1, the formula
- dF/F
is therefore a pullback of the invariant form.
Examples
- Exponential growthExponential growthExponential growth occurs when the growth rate of a mathematical function is proportional to the function's current value...
and exponential decay are processes with constant logarithmic derivative. - In mathematical financeMathematical financeMathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...
, the Greek λ is the logarithmic derivative of derivative price with respect to underlying price.