LOGARITHMIC DERIVATIVE Meaning and
Definition
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Logarithmic derivative refers to a mathematical concept used to measure the rate of change of a function relative to its own values. It is defined as the derivative of the natural logarithm of a function divided by the function itself.
In simpler terms, the logarithmic derivative allows us to quantify how a function changes as its input value varies. By evaluating the derivative of the logarithm of the function divided by the function itself, we obtain a value that can be used to describe the relative growth or decay of the function.
Formally, for a differentiable function f(x), the logarithmic derivative is represented as d/dx [ln(f(x))/f(x)]. This expression calculates the derivative of the natural logarithm of f(x), which measures the speed at which f(x) changes, and divides it by f(x) to normalize the result.
The logarithmic derivative provides a versatile tool in various mathematical and scientific fields. It is particularly useful in the study of exponential growth and decay, as well as in the analysis of complex functions and rates of change. For example, in number theory, the logarithmic derivative has applications in the study of prime numbers and their distribution.
By examining the logarithmic derivative, we can gain valuable insights into the behavior and characteristics of a function, helping us understand its growth or decline more comprehensively.
Etymology of LOGARITHMIC DERIVATIVE
The term "logarithmic derivative" is derived from the combination of two words - "logarithm" and "derivative".
The word "logarithm" comes from the Greek words "logos" meaning "ratio" or "proportion" and "arithmos" meaning "number". It was coined by the Scottish mathematician John Napier in the early 17th century. "Logarithm" refers to a mathematical function that relates to the exponent necessary to produce a given number within a specified base.
The term "derivative" comes from the Latin word "derivare" which means "to lead or draw off from". In mathematics, a derivative refers to the rate of change of a function with respect to its input variable.
