# Logarithmic Derivative

In mathematics , in particular calculus and complex analysis , the Logarithmic Derivative of a function f is defined by the formula

{\displaystyle {\frac {f'}{f}}}

where is the derivative of f . Intuitively, this is a subtle relative change in f ; That is, the subtle absolute change in f , i.e. f’, Incremented by the current value 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 natural logarithm of f . It follows the straight chain rule .

{\displaystyle {\frac {d}{dx}}\ln f(x)={\frac {1}{f(x)}}{\frac {df(x)}{dx}}}

## Core properties

Many properties of real logarithms also apply to logarithmic derivatives, even when the function does not take a value in a positive real. 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 Leibniz law for the derivative of a product

{\frac {(uv)'}{uv}}={\frac {u'v+uv'}{uv}}={\frac {u'}{u}}+{\frac {v'}{v}}.\!

Thus, it is true for any function that the log derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).

One consequence of this is that the log derivative of the inverse of a function is the negation of the log derivative of the function:

{\frac {(1 / u) '} {1 / u}} = {\frac {-u' /u ^ {2}} {1 /u}} = - {\frac {u '} {u} }, \!

Just as the logarithm of the inverse 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 between the logarithmic derivative of the dividend and the divisor:

{\frac {(u/v)'}{u/v}}={\frac {(u'v-uv')/v^{2}}{u/v}}={\frac {u'}{u}}-{\frac {v'}{v}},\!

Just as the logarithm of a quotient is the difference between the logarithm of the divisor and the divisor.

Generalizing in another direction, the logarithmic derivative of a power (with a constant real exponent) is the product of the logarithmic derivative of the exponent and the base:

{\frac {(u ^ {k}) '} {u ^ {k}}} = {\frac {ku ^ {k-1} u'} {u ^ {k}}} = k {\frac { u '} {u}}, \!

Just as the logarithm of a power is the product of the exponent and the logarithm of the base.

In short, both derivatives and logarithms have a product rule , a reciprocal rule , a quotient rule , and a power rule ( compare List of logarithmic identities ); Each pair of rules is related by means of a logarithmic derivative.

## Calculating Simple Derivatives Using Logarithmic Derivatives

Logarithmic derivatives can simplify the computation of derivatives requiring the product rule while producing similar results . The procedure is as follows: Let’s say ( x ) =  u ( x ) v ( x ) and we want to calculate ‘( x ) . Instead of computing it directly as =’ =  u’ v + v’ u , we calculate its logarithmic derivative. That is, we calculate:

{\frac {f'}{f}}={\frac {u'}{u}}+{\frac {v'}{v}}.

Multiplying by ‘ is calculate as :

{\displaystyle f'=f\cdot \left({\frac {u'}{u}}+{\frac {v'}{v}}\right).}

This technique is most useful when when is the product of a large number of factors. This technique makes it possible to calculate the logarithmic derivative of each factor, sum and multiply by to calculate ‘ .

## Integrating factors

The idea of ​​logarithmic derivatives is closely linked to the integrating factor method for first-order differential equations . In operator terms, write

{\displaystyle D={\frac {d}{dx}}}

And let M denote the operator of the multiplication by a given function G ( x ). Then

{\displaystyle M^{-1}DM}

( by product rule ) can be written as:

{\displaystyle D+M^{*}}

where is now denotes the multiplication operator by the logarithmic derivative M*

{\displaystyle {\frac {G'}{G}}}

In practice we are given an operator like

{\displaystyle D+F=L}

and want to solve the equations

{\displaystyle L(h)=f}

For the function h , f is given . It then reduces to solving

{\displaystyle {\frac {G'}{G}}=F}

whose solution is

{\displaystyle \exp \textstyle (\int F)}

With any indefinite integral of f .

## Complex analysis

The given formula can be applied more broadly; For example, if f ( z ) is a meromorphic function , it makes sense for all complex values ​​of z at which f has neither a zero nor a pole . Furthermore, the logarithmic derivative at a zero or a pole behaves in a way that is easily analyzed in terms of the special case. zn
With n an integer, n 0. The logarithmic derivative is thenn / z ;
And one can draw the general conclusion that for f meromorphic, the specificity of the logarithmic derivative of f are all simple poles, with residues n of order minus n , residues – n from a pole of order n . See logic theory . This information is often used in contour integration .

In the field of Nevanlinna theory , an important lemma states that, for example, the adjacency function of the logarithmic derivative with respect to the Nevanlinna characteristic of the original function is trivial .

m(r,h'/h)=S(r,h)=o(T(r,h))

## Multiplier group

Behind the use of the logarithmic derivative are two basic facts about GL1 , which is the multiplicative set of real numbers or other fields . difference operator

X{\frac {d}{dX}}

is invariant under ‘translation’ (replacing x by a constant for ax ). and differential form dx/x

Likewise it is immutable. For the function F to GL 1 , the formula df/f

Therefore there is a pullback of the irreversible form .

Example

• Exponential growth and exponential decay are processes with a constant logarithmic derivative.
• In mathematical finance , the Greek derivative with respect to the underlying price is the logarithmic derivative of the price.
• In numerical analysis , the condition number is a small relative change in output to a relative change in input, and thus is the ratio of logarithmic derivatives.