Logarithmic derivative

In analysis , the logarithmic derivative of a differentiable function that has no zeros is defined as the quotient of the derivative of a function and the function itself; formally ${\displaystyle \operatorname {L} (f)}$ ${\displaystyle f}$

${\displaystyle \operatorname {L} (f):={\frac {f'}{f}}.}$

In the same way, the term can also be defined for non-zero meromorphic functions (here no zeros need to be excluded because the quotient for meromorphic functions is well defined). For real functions with positive values, the logarithmic derivative according to the chain rule agrees with the derivative of the function ; therefore the name. So it applies ${\displaystyle f}$ ${\displaystyle \ln(f)}$

${\displaystyle (\ln f)'={\frac {f'}{f}}}$.

Calculation rules

The meaning of the term lies in the formula for the logarithmic derivative of a product:

${\displaystyle \operatorname {L} (f\cdot g)=\operatorname {L} (f)+\operatorname {L} (g)}$,

general

${\displaystyle \operatorname {L} (f_{1}\cdots f_{n})=\operatorname {L} (f_{1})+\ldots +\operatorname {L} (f_{n})}$.

As a modification of the product rule applies

${\displaystyle (fg)'=fg(\operatorname {L} (f)+\operatorname {L} (g))}$.

Applies analogously

${\displaystyle \operatorname {L} (1/f)=-\operatorname {L} (f)}$

and

${\displaystyle \operatorname {L} (f/g)=\operatorname {L} (f)-\operatorname {L} (g)}$.

For the logarithmic derivative of the power function one obtains approximately

${\displaystyle \operatorname {L} (f^{n})=n\cdot \operatorname {L} (f)}$.

These formulas follow from Leibniz's rule and therefore also apply in a more general context, for example in the (formal) derivation of polynomials or rational functions over any basic body .

Examples

The logarithmic derivative of functions can usually be determined with the normal differentiation rules.

${\displaystyle f}$	${\displaystyle \operatorname {L} (f)}$	Remarks
${\displaystyle n}$	${\displaystyle 0}$	${\displaystyle n\in \mathbb {R} \setminus \{0\}}$
${\displaystyle x^{n}}$	${\displaystyle {\frac {n}{x}}}$	${\displaystyle n\in \mathbb {R} }$
${\displaystyle e^{nx}}$	${\displaystyle n}$	${\displaystyle n\in \mathbb {R} }$
${\displaystyle \ln(x)}$	${\displaystyle {\frac {1}{x\ln(x)}}}$
${\displaystyle \sin(x)}$	${\displaystyle \cot(x)}$
${\displaystyle \cos(x)}$	${\displaystyle -\tan(x)}$
${\displaystyle \tan(x)}$	${\displaystyle {\frac {1}{\sin(x)\cos(x)}}}$
${\displaystyle \Gamma (z)}$	${\displaystyle \psi (z)}$	The logarithmic derivative of the gamma function is the digamma function .

Function theory

Let it be a meromorphic function with a zero of order or a pole of order at one point . Then as ${\displaystyle g(z)}$ ${\displaystyle n}$ ${\displaystyle -n}$ ${\displaystyle c\in \mathbb {C} }$ ${\displaystyle g(z)}$

${\displaystyle g(z)=(z-c)^{n}f(z)}$

with a writing in an environment of holomorphic function with . It applies ${\displaystyle c}$ ${\displaystyle f(z)}$ ${\displaystyle f(c)\neq 0}$

${\displaystyle L(g)={\frac {n}{z-c}}+L(f).}$

Wegen is in a neighborhood of holomorphic. The residual of at that point corresponds to the order of the zeros of at that point . This connection is in principle used by the argument . ${\displaystyle f(c)\neq 0}$ ${\displaystyle L(f)}$ ${\displaystyle c}$ ${\displaystyle L(g)}$ ${\displaystyle c}$ ${\displaystyle g}$ ${\displaystyle c}$

application

Can be represented as a function ${\displaystyle f}$

${\displaystyle f=k\cdot u^{a}\cdot v^{b}\cdot w^{c}\cdot \dots }$

with and as constants, the derivation is as follows ${\displaystyle k}$ ${\displaystyle a,b,c,\dots }$

${\displaystyle f'=f\cdot \left(a\cdot {\frac {u'}{u}}+b\cdot {\frac {v'}{v}}+c\cdot {\frac {w'}{w}}+\dots \right).}$

This circumstance can in practical applications such as hand account be used to some derivation rules compact summarize: This results, for example in the factors , , the product rule , by the factors , , the quotient rule and , the reciprocal rule . ${\displaystyle k=1}$ ${\displaystyle a=1}$ ${\displaystyle b=1}$ ${\displaystyle k=1}$ ${\displaystyle a=1}$ ${\displaystyle b=-1}$ ${\displaystyle k=1}$ ${\displaystyle a=-1}$

literature

• Richard P. Feynman, Michael A. Gottlieb, Ralph Leighton: Feynman's Tips on Physics: A Problem-Solving Supplement to the Feynman Lectures on Physics. Addison-Wesley, San Francisco, 2006, ISBN 0-8053-9063-4 , Chapters 1-4.