# Logarithmic derivative

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In mathematics, specifically in calculus and complex analysis, the **logarithmic derivative** of a function *f* is defined by the formula

where is the derivative of *f*. Intuitively, this is the infinitesimal relative change in *f*; that is, the infinitesimal absolute change in *f,* namely scaled 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*. This follows directly from the chain rule.

## 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 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 to this is that 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.

## Computing ordinary derivatives using logarithmic derivatives

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Logarithmic derivatives can simplify the computation of derivatives requiring the product rule. The procedure is as follows: Suppose that ƒ(*x*) = *u*(*x*)*v*(*x*) and that we wish to compute ƒ'(*x*). Instead of computing it directly, we compute its logarithmic derivative. That is, we compute:

Multiplying through by ƒ computes ƒ':

This technique is most useful when ƒ is a product of a large number of factors. This technique makes it possible to compute ƒ' 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 factor 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*^{−1}*DM*

can be written (by the product rule) 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 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

*z*^{n}

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 *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 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 .

## The multiplicative group

Behind the use of the logarithmic derivative lie two basic facts about *GL*_{1}, that is, the multiplicative group of real numbers or other field. The differential operator

is invariant under 'translation' (replacing *X* by *aX* for *a* constant). And the differential form

*dX/X*

is likewise invariant. For functions *F* into *GL*_{1}, the formula

*dF/F*

is therefore a *pullback* of the invariant form.

## Examples

- Exponential growth and exponential decay are processes with constant logarithmic derivative.
- In mathematical finance, the Greek
*λ*is the logarithmic derivative of derivative price with respect to underlying price. - 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.

## See also

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