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The logarithmic derivative is then / and one can draw the general conclusion that for f meromorphic, the singularities of the logarithmic derivative of f are all simple poles, with residue n from a zero of order n, residue −n from a pole of order n. See argument principle. This information is often exploited in contour integration.
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] () ′ = ′ ′ = () ′.
The derivative with a generalized functional argument f(x) is = ′ (). The quotient at the right hand side is called the logarithmic derivative of f. Computing f'(x) by means of the derivative of ln(f(x)) is known as logarithmic differentiation. [38]
In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: [1] [2] [3]
The natural logarithm of a number is its logarithm to the base of the mathematical constant e, which is an irrational and transcendental number approximately equal to 2.718 281 828 459. [1] The natural logarithm of x is generally written as ln x, log e x, or sometimes, if the base e is implicit, simply log x.
The identities of logarithms can be used to approximate large numbers. Note that log b (a) + log b (c) = log b (ac), where a, b, and c are arbitrary constants. Suppose that one wants to approximate the 44th Mersenne prime, 2 32,582,657 −1. To get the base-10 logarithm, we would multiply 32,582,657 by log 10 (2), getting 9,808,357.09543 ...
3.1 Proof from derivative definition and limit properties. 3.2 Proof using implicit differentiation. ... Taking the logarithmic derivative of both sides, ...
Logarithmic differentiation; Related rates; Taylor's theorem ... the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output ...