Logarithmic Differentiation

Overview

For a function

logarithmic differentiation typically begins by taking the natural logarithm, or the logarithm to the base e, on both sides, remembering to take absolute values

After implicit differentiation

Multiplication by y is then done to eliminate 1/y and leave only dy/dx on the left:

The method is used because the properties of logarithms provide avenues to quickly simplify complicated functions to be differentiated. These properties can be manipulated after the taking of natural logarithms on both sides and before the preliminary differentiation. The most commonly used logarithm laws:

$\log(ab) = \log(a) + \log(b), \qquad \log\left(\frac{a}{b}\right) = \log(a) - \log(b), \qquad \log(a^n) = n\log(a)$

Read more about this topic: Logarithmic Differentiation

Logarithmic Differentiation - Overview