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In mathematics, specifically in calculus and complex analysis, the logarithmic derivative of a function f is defined by the formula
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:^{[1]}
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Derivative of Logarithmic Functions

Derivatives of Exponential Functions & Logarithmic Differentiation Calculus lnx, e^2x, x^x, x^sinx

❖ Derivatives of Logarithmic Functions ❖

Differentiating logarithmic functions using log properties  AP Calculus AB  Khan Academy

Derivatives of Logarithmic and Exponential Functions
Transcription
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
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:
More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives 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:
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 , we compute its logarithmic derivative. That is, we compute:
Multiplying through by ƒ computes f′:
This technique is most useful when ƒ is a product of a large number of factors. This technique makes it possible to compute f′ by computing the logarithmic derivative of each factor, summing, and multiplying by f.
For example, we can compute the logarithmic derivative of to be .
Integrating factors
The logarithmic derivative idea is closely connected to the integrating factor method for firstorder differential equations. In operator terms, write
In practice we are given an operator such as
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, n ≠ 0. The logarithmic derivative is then
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 .^{[4]}^{[verification needed]}
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
Examples
 Exponential growth and exponential decay are processes with constant logarithmic derivative.^{[citation needed]}
 In mathematical finance, the Greek λ is the logarithmic derivative of derivative price with respect to underlying price.^{[citation needed]}
 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.^{[citation needed]}
See also
 Generalizations of the derivative – Fundamental construction of differential calculus
 Logarithmic differentiation – Method of mathematical differentiation
 Elasticity of a function
References
 ^ ^{a} ^{b} "Logarithmic derivative  Encyclopedia of Mathematics". encyclopediaofmath.org. 7 December 2012. Retrieved 12 August 2021.
 ^ Gonzalez, Mario (19910924). Classical Complex Analysis. CRC Press. ISBN 9780824784157.
 ^ "Logarithmic residue  Encyclopedia of Mathematics". encyclopediaofmath.org. 7 June 2020. Retrieved 20210812.
 ^ Zhang, Guanhou (19930101). Theory of Entire and Meromorphic Functions: Deficient and Asymptotic Values and Singular Directions. American Mathematical Soc. p. 18. ISBN 9780821887646. Retrieved 12 August 2021.