Editing Logarithmic derivative (section)

==Basic properties==
Many properties of the real logarithm also apply to the logarithmic derivative, even when the function does ''not'' take values in the positive reals. For example, since the logarithm of a product is the sum of the logarithms of the factors, we have
<math display="block"> (\log uv)' = (\log u + \log v)' = (\log u)' + (\log v)' . </math>
So for positive-real-valued functions, the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors. But we can also use the [[General Leibniz rule|Leibniz law]] for the derivative of a product to get
<math display="block"> \frac{(uv)'}{uv} = \frac{u'v + uv'}{uv} = \frac{u'}{u} + \frac{v'}{v} . </math>
Thus, it is true for ''any'' function that the logarithmic derivative of a product is the sum of the logarithmic derivatives of the factors (when they are defined).

A [[corollary]] to this is that the logarithmic derivative of the reciprocal of a function is the negation of the logarithmic derivative of the function:
<math display="block"> \frac{(1/u)'}{1/u} = \frac{-u'/u^{2}}{1/u} = -\frac{u'}{u} , </math>
just as the logarithm of the reciprocal of a positive real number is the negation of the logarithm of the number.{{Citation needed|date=August 2021}}

More generally, the logarithmic derivative of a quotient is the difference of the logarithmic derivatives of the dividend and the divisor:
<math display="block"> \frac{(u/v)'}{u/v} = \frac{(u'v - uv')/v^{2}}{u/v} = \frac{u'}{u} - \frac{v'}{v} , </math>
just as the logarithm of a quotient is the difference of the logarithms of the dividend and the divisor.

Generalising in another direction, the logarithmic derivative of a power (with constant real exponent) is the product of the exponent and the logarithmic derivative of the base:
<math display="block"> \frac{(u^{k})'}{u^{k}} = \frac {ku^{k-1}u'}{u^{k}} = k \frac{u'}{u} , </math>
just as the logarithm of a power is the product of the exponent and the logarithm of the base.

In summary, both derivatives and logarithms have a [[product rule]], a [[reciprocal rule]], a [[quotient rule]], and a [[power rule]] (compare the [[list of logarithmic identities]]); each pair of rules is related through the logarithmic derivative.