In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function where is the derivative of . Intuitively, this is the infinitesimal relative change in ; that is, the infinitesimal absolute change in , namely scaled by the current value of .

When is a function of a real variable , and takes real, strictly positive values, this is equal to the derivative of , or the natural logarithm of . 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 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 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 just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.

just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.

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

Finally, the logarithmic derivative of the exponential function of is just the derivative of just as the logarithm of the exponential function of a function is just the original function.

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

Logarithmic derivatives can simplify the computation of derivatives requiring the product rule while producing the same result. The procedure is as follows: Suppose that and that we wish to compute . Instead of computing it directly as

Multiplying through by computes

This technique is most useful when 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 .

Integrating factors

The logarithmic derivative idea is closely connected to the integrating factor method for first-order differential equations. In operator and let M denote the operator of multiplication by some given function G(x can be written (by the product rule

for the function h, given 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

with n an integer, 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.

The multiplicative group

Behind the use of the logarithmic derivative lie two basic facts about GL1, that is, the multiplicative group of real numbers or other field. The differential operator is invariant under dilation (replacing X by aX for a constant). And the differential form is likewise invariant. For functions F into GL1 is therefore a pullback of the invariant form.

Examples

• Exponential growth and exponential decay are processes with constant logarithmic derivative.

• In mathematical finance, the Greekλ 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.

• The digamma function, and by extension the polygamma function, is defined in terms of the logarithmic derivative of the gamma function.

See also

• Elasticity of a function

• Product integral

References