Logarithmic Derivative - Computing Ordinary Derivatives Using Logarithmic Derivatives

Computing Ordinary Derivatives Using Logarithmic Derivatives

Logarithmic derivatives can simplify the computation of derivatives requiring the product rule. The procedure is as follows: Suppose that ƒ(x) = u(x)v(x) and that we wish to compute ƒ'(x). Instead of computing it directly, we compute its logarithmic derivative. That is, we compute:

Multiplying through by ƒ computes ƒ':

This technique is most useful when ƒ is a product of a large number of factors. This technique makes it possible to compute ƒ' by computing the logarithmic derivative of each factor, summing, and multiplying by ƒ.