The logarithmic derivative of a function is defined as the derivative of the logarithm of a function. For example, the digamma function is defined as the logarithmic derivative of the gamma function,

Zwillinger, D. (Ed.). "Logarithmic Derivative." §6.11.8 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, p. 496, 1995.

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Weisstein, Eric W. "Logarithmic Derivative." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/LogarithmicDerivative.html