This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Logarithmic derivative" – news ·newspapers ·books ·scholar ·JSTOR(August 2021) (Learn how and when to remove this message)

Part of a series of articles about
Calculus
${\displaystyle {\int }_a^b{f}^{\prime }(t)dt=f(b)-f(a)}$
Fundamental theorem Limits Continuity Rolle's theorem Mean value theorem Inverse function theorem
Differential Definitions Derivative (generalizations) Differential infinitesimal of a function total Concepts Differentiation notation Second derivative Implicit differentiation Logarithmic differentiation Related rates Taylor's theorem Rules and identities Sum Product Chain Power Quotient L'Hôpital's rule Inverse General Leibniz Faà di Bruno's formula Reynolds
Integral Lists of integrals Integral transform Leibniz integral rule Definitions Antiderivative Integral (improper) Riemann integral Lebesgue integration Contour integration Integral of inverse functions Integration by Parts Discs Cylindrical shells Substitution (trigonometric,tangent half-angle,Euler) Euler's formula Partial fractions (Heaviside's method) Changing order Reduction formulae Differentiating under the integral sign Risch algorithm
Series Geometric (arithmetico-geometric) Harmonic Alternating Power Binomial Taylor Convergence tests Summand limit (term test) Ratio Root Integral Direct comparison Limit comparison Alternating series Cauchy condensation Dirichlet Abel
Vector Gradient Divergence Curl Laplacian Directional derivative Identities Theorems Gradient Green's Stokes' Divergence Generalized Stokes Helmholtz decomposition
Multivariable Formalisms Matrix Tensor Exterior Geometric Definitions Partial derivative Multiple integral Line integral Surface integral Volume integral Jacobian Hessian
Advanced Calculus on Euclidean space Generalized functions Limit of distributions
Specialized Fractional Malliavin Stochastic Variations
Miscellanea Precalculus History Glossary List of topics Integration Bee Mathematical analysis Nonstandard analysis
v t e

Inmathematics, specifically incalculus andcomplex analysis, thelogarithmic derivative of afunctionf is defined by the formula$${\displaystyle \frac{f^{\prime }}{f}}$$wheref′ is thederivative off.^[1] Intuitively, this is the infinitesimalrelative change inf; that is, the infinitesimal absolute change inf, namelyf′ scaled by the current value off.

Whenf is a functionf(x) of a real variablex, and takesreal, strictlypositive values, this is equal to the derivative oflnf(x), or thenatural logarithm off. This follows directly from thechain rule:^[1]$${\displaystyle \frac{d}{dx}\ln f(x)=\frac{1}{f(x)}\frac{df(x)}{dx}}$$

Many properties of the real logarithm also apply to the logarithmic derivative, even when the function doesnot 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$${\displaystyle (\log uv{)}^{\prime }=(\log u+\log v{)}^{\prime }=(\log u{)}^{\prime }+(\log v{)}^{\prime }.}$$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 theLeibniz law for the derivative of a product to get$${\displaystyle \frac{(uv{)}^{\prime }}{uv}=\frac{u^{\prime }v+u{v}^{\prime }}{uv}=\frac{u^{\prime }}{u}+\frac{v^{\prime }}{v}.}$$Thus, it is true forany function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).

Acorollary to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:$${\displaystyle \frac{(1/u{)}^{\prime }}{1/u}=\frac{-u^{\prime }/u^2}{1/u}=-\frac{u^{\prime }}{u},}$$just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.^{[citation needed]}

More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor:$${\displaystyle \frac{(u/v{)}^{\prime }}{u/v}=\frac{(u^{\prime }v-u{v}^{\prime })/v^2}{u/v}=\frac{u^{\prime }}{u}-\frac{v^{\prime }}{v},}$$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:$${\displaystyle \frac{(u^k{)}^{\prime }}{u^k}=\frac{k{u}^{k-1}{u}^{\prime }}{u^k}=k\frac{u^{\prime }}{u},}$$just as the logarithm of a power is the product of the exponent and the logarithm of the base.

Finally, the logarithmic derivative of theexponential function ofu is just the derivative ofu:$${\displaystyle \frac{(e^u{)}^{\prime }}{e^u}=\frac{u^{\prime }{e}^u}{e^u}=u^{\prime },}$$just as the logarithm of the exponential function of a function is just the original function.

In summary, both derivatives and logarithms have aproduct rule, areciprocal rule, aquotient rule, and apower rule (compare thelist of logarithmic identities); each pair of rules is related through the logarithmic derivative.

Logarithmic derivatives can simplify the computation of derivatives requiring theproduct rule while producing the same result. The procedure is as follows: Suppose thatf(x) =u(x)v(x) and that we wish to computef′(x). Instead of computing it directly asf′ =u′v +v′u, we compute its logarithmic derivative. That is, we compute:$${\displaystyle \frac{f^{\prime }}{f}=\frac{u^{\prime }}{u}+\frac{v^{\prime }}{v}.}$$

Multiplying through byf computesf′:$${\displaystyle f^{\prime }=f\cdot \left(\frac{u^{\prime }}{u}+\frac{v^{\prime }}{v}\right).}$$

This technique is most useful whenf is a product of a large number of factors. This technique makes it possible to computef′ by computing the logarithmic derivative of each factor, summing, and multiplying byf.

For example, we can compute the logarithmic derivative of$${\displaystyle (e^x{)}^2(x-2{)}^3(x-3)(x-1{)}^{-1}}$$to be$${\displaystyle 2+\frac{3}{x-2}+\frac{1}{x-3}-\frac{1}{x-1}.}$$

The logarithmic derivative idea is closely connected to theintegrating factor method forfirst-order differential equations. Inoperator terms, write$${\displaystyle D=\frac{d}{dx}}$$ and letM denote the operator of multiplication by some given functionG(x). Then$${\displaystyle M^{-1}DM}$$ can be written (by theproduct rule) as$${\displaystyle D+M^{\ast }}$$ where${\displaystyle M^{\ast }}$ now denotes the multiplication operator by the logarithmic derivative$${\displaystyle \frac{G^{\prime }}{G}}$$

In practice we are given an operator such as$${\displaystyle D+F=L}$$and wish to solve equations$${\displaystyle L(h)=f}$$for the functionh, givenf. This then reduces to solving$${\displaystyle \frac{G^{\prime }}{G}=F}$$which has as solution$${\displaystyle \exp (\int F)}$$with anyindefinite integral ofF.^{[citation needed]}

The formula as given can be applied more widely; for example iff(z) is ameromorphic function, it makes sense at all complex values ofz at whichf has neither azero 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

withn an integer,n ≠ 0. The logarithmic derivative is then$${\displaystyle n/z}$$and one can draw the general conclusion that forf meromorphic, the singularities of the logarithmic derivative off are allsimple poles, withresiduen from a zero of ordern, residue −n from a pole of ordern. Seeargument principle. This information is often exploited incontour integration.^{[2][3][verification needed]}

In the field ofNevanlinna 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${\displaystyle m(r,h^{\prime }/h)=S(r,h)=o(T(r,h))}$.^{[4][verification needed]}

Behind the use of the logarithmic derivative lie two basic facts aboutGL₁, that is, the multiplicative group ofreal numbers or otherfield. Thedifferential operator$${\displaystyle X\frac{d}{dX}}$$isinvariant under dilation (replacingX byaX fora constant). And thedifferential form$${\displaystyle \frac{dx}{X}}$$ is likewise invariant. For functionsF into GL₁, the formula$${\displaystyle \frac{dF}{F}}$$ is therefore apullback of the invariant form.^{[citation needed]}

• Exponential growth andexponential decay are processes with constant logarithmic derivative.^{[citation needed]}
• Inmathematical finance, theGreek λ is the logarithmic derivative of derivative price with respect to underlying price.^{[citation needed]}
• Innumerical analysis, thecondition number is the infinitesimal relative change in the output for a relative change in the input, and is thus a ratio of logarithmic derivatives.^{[citation needed]}
• Thedigamma function, and by extension thepolygamma function, is defined in terms of the logarithmic derivative of thegamma function.

• Generalizations of the derivative – Fundamental construction of differential calculus
• Logarithmic differentiation – Method of mathematical differentiation
• Elasticity of a function
• Product integral

• Differential calculus
• Complex analysis

• Articles with short description
• Short description is different from Wikidata
• Articles needing additional references from August 2021
• All articles needing additional references
• Pages using sidebar with the child parameter
• All articles with unsourced statements
• Articles with unsourced statements from August 2021
• All pages needing factual verification
• Wikipedia articles needing factual verification from August 2021