The logarithmic derivative of a function is a useful concept in calculus, particularly in the context of growth rates and relative changes. For a differentiable function \( f(x) \), the logarithmic derivative is defined as the derivative of the natural logarithm of the function.
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The logarithmic derivative of a differentiable function is wherever . For a nonvanishing holomorphic function, a local holomorphic logarithm differentiates to this expression even when no global logarithm exists. If a meromorphic function has local factorization with , differentiation gives . Its residue is therefore , positive for a zero and negative for a pole; this explains the multiplicity count in the argument principle. For , the Euler product for the Riemann zeta function gives