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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]
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] () ′ = ′ ′ = () ′.
7.2 Derivatives of logarithmic functions. 7.3 Integral definition. 7.4 Riemann Sum. 7.5 Series representation. 7.6 Harmonic number difference. 7.6.1 Harmonic limit ...
Because log(x) is the sum of the terms of the form log(1 + 2 −k) corresponding to those k for which the factor 1 + 2 −k was included in the product P, log(x) may be computed by simple addition, using a table of log(1 + 2 −k) for all k. Any base may be used for the logarithm table. [53]
How to establish this derivative of the natural logarithm depends on how it is defined firsthand. If the natural logarithm is defined as the integral ln x = ∫ 1 x 1 t d t , {\displaystyle \ln x=\int _{1}^{x}{\frac {1}{t}}\,dt,} then the derivative immediately follows from the first part of the fundamental theorem of calculus .
The derivatives in the table above are for when the range of the inverse secant is [,] and when the range of the inverse cosecant is [,]. It is common to additionally define an inverse tangent function with two arguments , arctan ( y , x ) {\textstyle \arctan(y,x)} .
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The derivative of the complex logarithm [ edit ] Each branch L ( z ) {\displaystyle \operatorname {L} (z)} of log z {\displaystyle \log z} on an open set U {\displaystyle U} is the inverse of a restriction of the exponential function, namely the restriction to the image L ( U ) {\displaystyle \operatorname {L} (U)} .