Logarithmic derivative

Analysis of the logarithmic derivative of a differentiable function which has no zeros, as the quotient of a function and its derivative defined; formally

For real functions with positive values ​​he agrees after the chain rule to the derivative of the function; hence the name. It is therefore

For holomorphic or meromorphic functions but the logarithmic derivative can also be formed, although the complex logarithm can not be defined at all.

Calculation rules

The meaning of the term is, in the formula for the logarithmic derivative of the product:

Generally

As a modification to the product rule so

Analogously,

And

For the logarithmic derivative of the power function is obtained as

These formulas follow from the Leibniz rule and are thus also in a more general context, for example in the (formal) derivative of polynomials or rational functions over an arbitrary base.

Examples

The logarithmic derivative of functions can be usually determined with the normal rules of differentiation.

Application

Can be represented as a function

And with a constant, the result is the derivative to

This circumstance can be used in practical applications such as hand calculation to summarize some compact derivation rules: For results, for example at the factors, the product rule, the factors, the quotient rule and with that Reziprokenregel.