 Logarithmic derivative

The logarithmic derivative of a differentiable function $f$ is $f^{\prime }/f$ wherever $f\ne 0$. 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 $f(z)=(z-a{)}^{m}h(z)$ with $h(a)\ne 0$, differentiation gives $f^{\prime }/f=m/(z-a)+h^{\prime }/h$. Its residue is therefore $m$, positive for a zero and negative for a pole; this explains the multiplicity count in the argument principle. For $\Re s>1$, the Euler product for the Riemann zeta function gives

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.