Logarithmic differentiation

Short description: Method of mathematical differentiation

Part of a series of articles about

Calculus

* Fundamental theorem

* Limits of functions
* Continuity

* Mean value theorem
* Rolle's theorem

Differential

|  Definitions

* Derivative (generalizations)

* Differential
* infinitesimal
* of a function
* total

Concepts

* Differentiation notation
* Second derivative
* Third derivative
* Change of variables
* Implicit differentiation
* Related rates
* Taylor's theorem

Rules and identities

* Sum
* Product
* Chain
* Power
* Quotient
* Inverse
* General Leibniz
* Faà di Bruno's formula

Integral

* Lists of integrals

Definitions

* Antiderivative
* Integral (improper)
* Riemann integral
* Lebesgue integration
* Contour integration

Integration by

* Parts
* Discs
* Cylindrical shells
* Substitution (trigonometric)
* Partial fractions
* Order
* Reduction formulae

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

* Divergence
* Gradient
* Green's
* Kelvin–Stokes
* Stokes

Multivariable

|  Formalisms

* Matrix
* Tensor
* Exterior
* Geometric

Definitions

* Partial derivative
* Multiple integral
* Line integral
* Surface integral
* Volume integral
* Jacobian
* Hessian

Specialized

* Fractional
* Malliavin
* Stochastic
* Variations

Glossary of calculus

* Glossary of calculus

* v
* t
* e

In calculus, logarithmic differentiation or differentiation by taking logarithms is a method used to differentiate functions by employing the logarithmic derivative of a function f,[1] (ln⁡f)′=f′f⟹f′=f⋅(ln⁡f)′.

The technique is often performed in cases where it is easier to differentiate the logarithm of a function rather than the function itself. This usually occurs in cases where the function of interest is composed of a product of a number of parts, so that a logarithmic transformation will turn it into a sum of separate parts (which is much easier to differentiate). It can also be useful when applied to functions raised to the power of variables or functions. Logarithmic differentiation relies on the chain rule as well as properties of logarithms (in particular, the natural logarithm, or the logarithm to the base e) to transform products into sums and divisions into subtractions.[2][3] The principle can be implemented, at least in part, in the differentiation of almost all differentiable functions, providing that these functions are non-zero.

## Contents

* 1 Overview
* 1.1 Higher order derivatives
* 2 Applications
* 2.1 Products
* 2.2 Quotients
* 2.3 Functional exponents
* 2.3.1 General case
* 3 See also
* 4 Notes

## Overview

The method is used because the properties of logarithms provide avenues to quickly simplify complicated functions to be differentiated.[4] These properties can be manipulated after the taking of natural logarithms on both sides and before the preliminary differentiation. The most commonly used logarithm laws are[3] ln⁡(ab)=ln⁡(a)+ln⁡(b),ln⁡(ab)=ln⁡(a)−ln⁡(b),ln⁡(an)=nln⁡(a).

### Higher order derivatives

Using Faà di Bruno's formula, the n-th order logarithmic derivative is, dndxnln⁡f(x)=∑m1+2m2+⋯+nmn=nn!m1!m2!⋯mn!⋅(−1)m1+⋯+mn−1(m1+⋯+mn−1)!f(x)m1+⋯+mn⋅∏j=1n(f(j)(x)j!)mj. Using this, the first four derivatives are, d2dx2ln⁡f(x)=f″(x)f(x)−(f′(x)f(x))2d3dx3ln⁡f(x)=f(3)(x)f(x)−3f′(x)f″(x)f(x)2+2(f′(x)f(x))3d4dx4ln⁡f(x)=f(4)(x)f(x)−4f′(x)f(3)(x)f(x)2−3(f″(x)f(x))2+12f′(x)2f″(x)f(x)3−6(f′(x)f(x))4

## Applications

### Products

Main article: Product rule

A natural logarithm is applied to a product of two functions f(x)=g(x)h(x) to transform the product into a sum ln⁡(f(x))=ln⁡(g(x)h(x))=ln⁡(g(x))+ln⁡(h(x)). Differentiating by applying the chain and the sum rules yields f′(x)f(x)=g′(x)g(x)+h′(x)h(x), and, after rearranging, yields[5] f′(x)=f(x)×{g′(x)g(x)+h′(x)h(x)}=g(x)h(x)×{g′(x)g(x)+h′(x)h(x)}=g′(x)h(x)+g(x)h′(x), which is the product rule for derivatives.

### Quotients

Main article: Quotient rule

A natural logarithm is applied to a quotient of two functions f(x)=g(x)h(x) to transform the division into a subtraction ln⁡(f(x))=ln⁡(g(x)h(x))=ln⁡(g(x))−ln⁡(h(x)) Differentiating by applying the chain and the sum rules yields f′(x)f(x)=g′(x)g(x)−h′(x)h(x), and, after rearranging, yields f′(x)=f(x)×{g′(x)g(x)−h′(x)h(x)}=g(x)h(x)×{g′(x)g(x)−h′(x)h(x)}=g′(x)h(x)−g(x)h′(x)h(x)2,

which is the quotient rule for derivatives.

### Functional exponents

For a function of the form f(x)=g(x)h(x) the natural logarithm transforms the exponentiation into a product ln⁡(f(x))=ln⁡(g(x)h(x))=h(x)ln⁡(g(x)) Differentiating by applying the chain and the product rules yields f′(x)f(x)=h′(x)ln⁡(g(x))+h(x)g′(x)g(x), and, after rearranging, yields f′(x)=f(x)×{h′(x)ln⁡(g(x))+h(x)g′(x)g(x)}=g(x)h(x)×{h′(x)ln⁡(g(x))+h(x)g′(x)g(x)}. The same result can be obtained by rewriting f in terms of exp and applying the chain rule.

#### General case

Using capital pi notation, let f(x)=∏i(fi(x))αi(x) be a finite product of functions with functional exponents.

The application of natural logarithms results in (with capital sigma notation) ln⁡(f(x))=∑iαi(x)⋅ln⁡(fi(x)), and after differentiation, f′(x)f(x)=∑i[αi′(x)⋅ln⁡(fi(x))+αi(x)⋅fi′(x)fi(x)]. Rearrange to get the derivative of the original function, f′(x)=∏i(fi(x))αi(x)⏞f(x)×∑i{αi′(x)⋅ln⁡(fi(x))+αi(x)⋅fi′(x)fi(x)}⏞[ln⁡(f(x))]′.

## See also

* Darboux derivative
* Generalizations of the derivative - Fundamental construction of differential calculus
* Lie group - Group that is also a differentiable manifold with group operations that are smooth
* List of logarithmic identities - None
* Maurer–Cartan form - Mathematical concept

## Notes

1. ↑ Krantz, Steven G. (2003). Calculus demystified. McGraw-Hill Professional. pp. 170. ISBN 0-07-139308-0.
2. ↑ N.P. Bali (2005). Golden Differential Calculus. Firewall Media. pp. 282. ISBN 81-7008-152-1.
3. ↑ 3.0 3.1 Bird, John (2006). Higher Engineering Mathematics. Newnes. pp. 324. ISBN 0-7506-8152-7.
4. ↑ Blank, Brian E. (2006). Calculus, single variable. Springer. pp. 457. ISBN 1-931914-59-1.
5. ↑ Williamson, Benjamin (2008). An Elementary Treatise on the Differential Calculus. BiblioBazaar, LLC. pp. 25–26. ISBN 978-0-559-47577-1.

* v
* t
* e

Calculus

Precalculus|

* Binomial theorem
* Concave function
* Continuous function
* Factorial
* Finite difference
* Free variables and bound variables
* Fundamental theorem
* Graph of a function
* Linear function
* Mean value theorem
* Radian
* Rolle's theorem
* Secant
* Slope
* Tangent

Limits|

* Indeterminate form
* Limit of a function
* One-sided limit
* Limit of a sequence
* Orders of approximation
* (ε, δ)-definition of limit

Differential calculus|

* Chain rule
* Derivative
* Differential
* Differential equation
* Differential operator
* Implicit differentiation
* Inverse functions and differentiation
* L'Hôpital's rule
* Leibniz's rule
* Logarithmic derivative
* Mean value theorem
* Newton's method
* Notation
* Leibniz's notation
* Newton's notation
* Product rule
* Quotient rule
* Regiomontanus' angle maximization problem
* Related rates
* Simplest rules
* Constant factor rule in differentiation
* Linearity of differentiation
* Power rule
* Sum rule in differentiation
* Stationary point
* First derivative test
* Second derivative test
* Extreme value theorem
* Maxima and minima
* Taylor's theorem

Integral calculus|

* Antiderivative
* Arc length
* Constant of integration
* Differentiation under the integral sign
* Fundamental theorem of calculus
* Integral of secant cubed
* Integral of the secant function
* Integration by parts
* Integration by substitution
* Tangent half-angle substitution
* Partial fractions in integration
* Quadratic integral
* Proof that 22/7 exceeds π
* Simplest rules
* Constant factor rule in integration
* Linearity of integration
* Sum rule in integration
* Trapezoidal rule
* Trigonometric substitution

Vector calculus|

* Curl
* Directional derivative
* Divergence
* Divergence theorem
* Gradient
* Gradient theorem
* Green's theorem
* Laplacian
* Stokes' theorem

Multivariable calculus|

* Curvature
* Disc integration
* Divergence theorem
* Exterior
* Gabriel's Horn
* Geometric
* Hessian matrix
* Jacobian matrix and determinant
* Line integral
* Matrix
* Multiple integral
* Partial derivative
* Shell integration
* Surface integral
* Tensor
* Volume integral

Series|

* Abel's test
* Alternating
* Alternating series test
* Arithmetico-geometric sequence
* Binomial
* Cauchy condensation test
* Direct comparison test
* Dirichlet's test
* Euler–Maclaurin formula
* Fourier
* Geometric
* Harmonic
* Infinite
* Integral test for convergence
* Limit comparison test
* Maclaurin
* Power
* Ratio test
* Root test
* Taylor
* Term test

Special functions and numbers|

* Bernoulli numbers
* e (mathematical constant)
* Exponential function
* Natural logarithm
* Stirling's approximation

History of calculus|

* Adequality
* Brook Taylor
* Colin Maclaurin
* Generality of algebra
* Gottfried Wilhelm Leibniz
* Infinitesimal
* Infinitesimal calculus
* Isaac Newton
* Fluxion
* Law of Continuity
* Leonhard Euler
* Method of Fluxions
* The Method of Mechanical Theorems

Lists|

* Differentiation rules
* List of integrals of exponential functions
* List of integrals of hyperbolic functions
* List of integrals of inverse hyperbolic functions
* List of integrals of inverse trigonometric functions
* List of integrals of irrational functions
* List of integrals of logarithmic functions
* List of integrals of rational functions
* List of integrals of trigonometric functions
* List of limits
* List of mathematical symbols
* Lists of integrals

0.00

(0 votes)

Original source: https://en.wikipedia.org/wiki/Logarithmic differentiation. Read more

|

Retrieved from "https://handwiki.org/wiki/index.php?title=Logarithmic_differentiation&oldid=185069"
*[v]: View this template
*[t]: Discuss this template
*[e]: Edit this template