logarithmic derivative

Noun

 * 1)  Given a real or complex function $$f$$, the ratio of the value of the derivative to the value of the function, $$\frac {f'} f$$, regarded as a function.

Usage notes

 * The logarithmic derivative can be interpreted intuitively as the infinitesimal relative change in $$f$$ at any given point.
 * If $$f(x)$$ is a differentiable function of a real variable and takes only positive values (so that $$\ln f(x)$$ is defined), the chain rule applies and the logarithmic derivative is equal to the derivative of the logarithm: $$\textstyle\frac{f'(x)}{f(x)}= \left ( \ln f(x) \right )'$$.
 * The definition above is more broadly applicable: for $$f(z)$$ a function of a complex variable, its logarithmic derivative will be computable so long as $$f(z)\ne 0$$ and $$f'(z)$$ is defined.