Vipul: Created page with "==Definition== ===At a point=== Suppose $f$ is a function and $x_0$ is a point in the interior of the domain of $f$ such that the deri..."

2014-05-01T14:30:40Z

Created page with "==Definition== ===At a point=== Suppose <math>f</math> is a function and <math>x_0</math> is a point in the interior of the domain of <math>f</math> such that the deri..."

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==Definition==

===At a point===

Suppose <math>f</math> is a [[function]] and <math>x_0</math> is a point in the interior of the domain of <math>f</math> such that the [[derivative]] <math>f'(x_0)</math> exists, <math>x_0 \ne 0</math>, ''and'' <math>f(x_0) \ne 0</math> (note that <math>f'(x_0)</math> may be zero or nonzero). The '''logarithmic derivative''' of <math>f</math> at <math>x_0</math> is defined as:

<math>\! \frac{x_0f'(x_0)}{f(x_0)}</math>

Alternatively, it can be defined as:

<math>\! \frac{d(\ln|f(x)|}{d(\ln|x|)}|_{x = x_0}</math>

===As a function===

Suppose <math>f</math> is a [[function]] of one variable. The '''logarithmic derivative''' of <math>f</math> is the function:

<math>\! x \mapsto \frac{xf'(x)}{f(x)}</math>

Alternatively, it is the function:

<math>\! x \mapsto \frac{d(\ln|f(x)|}{d(\ln|x|)}</math>

The [[domain]] of this function is precisely the set of points <math>x</math> for which <math>f(x) \ne 0</math> and the [[derivative]] <math>f'</math> exists.

{{specific point generic point confusion}}

{{division by zero}}

Relative logarithmic derivative - Revision history

Vipul: Created page with "==Definition== ===At a point=== Suppose f is a function and x_0 is a point in the interior of the domain of f such that the deri..."

Vipul: Created page with "==Definition== ===At a point=== Suppose $f$ is a function and $x_0$ is a point in the interior of the domain of $f$ such that the deri..."