logarithmic derivative
التعريفات والمعاني
== English ==
=== Pronunciation ===
=== Noun ===
logarithmic derivative (plural logarithmic derivatives)
(calculus, mathematical analysis) Given a real or complex function
f
{\displaystyle f}
, the ratio of the value of the derivative to the value of the function,
f
′
f
{\displaystyle {\frac {f'}{f}}}
, regarded as a function.
==== Usage notes ====
The logarithmic derivative can be interpreted intuitively as the infinitesimal relative change in
f
{\displaystyle f}
at any given point.
If
f
(
x
)
{\displaystyle f(x)}
is a differentiable function of a real variable and takes only positive values (so that
ln
f
(
x
)
{\displaystyle \ln f(x)}
is defined), the chain rule applies and the logarithmic derivative is equal to the derivative of the logarithm:
f
′
(
x
)
f
(
x
)
=
(
ln
f
(
x
)
)
′
{\displaystyle \textstyle {\frac {f'(x)}{f(x)}}=\left(\ln f(x)\right)'}
.
The definition above is more broadly applicable: for
f
(
z
)
{\displaystyle f(z)}
a function of a complex variable, its logarithmic derivative will be computable so long as
f
(
z
)
≠
0
{\displaystyle f(z)\neq 0}
and
f
′
(
z
)
{\displaystyle f'(z)}
is defined.
==== Translations ====
=== Further reading ===
Logarithmic differentiation on Wikipedia.Wikipedia