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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: (/) ′ / = ′ / / = ′, just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.
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 logarithmic derivative is another way of stating the rule for differentiating the logarithm of a function (using the chain rule): () ′ = ′, wherever is positive. Logarithmic differentiation is a technique which uses logarithms and its differentiation rules to simplify certain expressions before actually applying the derivative.
The logarithmic derivative provides a simpler expression of the last form, as well as a direct proof that does not involve any recursion. The logarithmic derivative of a function f , denoted here Logder( f ) , is the derivative of the logarithm of the function.
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 ...
The digamma function (), visualized using domain coloring Plots of the digamma and the next three polygamma functions along the real line (they are real-valued on the real line) In mathematics, the digamma function is defined as the logarithmic derivative of the gamma function: [1] [2] [3]
2.1 Proof from derivative definition and ... , which justifies taking the absolute value of the functions for logarithmic differentiation. ...
The logarithmic derivative of the gamma function is called the digamma function; higher derivatives are the polygamma functions. The analog of the gamma function over a finite field or a finite ring is the Gaussian sums, a type of exponential sum. The reciprocal gamma function is an entire function and has been studied as a specific topic.