Editing Natural logarithm (section)

== Derivative ==

The [[derivative]] of the natural logarithm as a [[real-valued function]] on the positive reals is given by<ref name=":1" />
<math display="block">\frac{d}{dx} \ln x = \frac{1}{x}.</math>

How to establish this derivative of the natural logarithm depends on how it is defined firsthand.  If the natural logarithm is defined as the integral
<math display="block">\ln x = \int_1^x \frac{1}{t}\,dt,</math>
then the derivative immediately follows from the first part of the [[Fundamental theorem of calculus#First part|fundamental theorem of calculus]].

On the other hand, if the natural logarithm is defined as the inverse of the (natural) exponential function, then the derivative (for {{math|''x'' > 0}}) can be found by using the properties of the logarithm and a definition of the exponential function.

From the definition of the number <math>e = \lim_{u\to 0}(1+u)^{1/u},</math> the exponential function can be defined as
<math display="block">e^x = \lim_{u\to 0} (1+u)^{x/u} = \lim_{h\to 0}(1 + hx)^{1/h} , </math> where <math>u=hx, h=\frac{u}{x}.</math>

The derivative can then be found from first principles.
<math display="block">\begin{align}
   \frac{d}{dx} \ln x &= \lim_{h\to 0} \frac{\ln(x+h) - \ln x}{h} \\
   &= \lim_{h\to 0}\left[ \frac{1}{h} \ln\left(\frac{x+h}{x}\right)\right] \\
   &= \lim_{h\to 0}\left[ \ln\left(1 + \frac{h}{x}\right)^{\frac{1}{h}}\right]\quad &&\text{all above for logarithmic properties}\\
   &= \ln \left[ \lim_{h\to 0}\left(1 + \frac{h}{x}\right)^{\frac{1}{h}}\right]\quad &&\text{for continuity of the logarithm} \\
   &= \ln e^{1/x} \quad &&\text{for the definition of } e^x = \lim_{h\to 0}(1 + hx)^{1/h}\\
   &= \frac{1}{x} \quad &&\text{for the definition of the ln as inverse function.}
 \end{align}</math>

Also, we have:
<math display="block">\frac{d}{dx} \ln ax = \frac{d}{dx} (\ln a + \ln x) = \frac{d}{dx} \ln a +\frac{d}{dx} \ln x = \frac{1}{x}.</math>

so, unlike its inverse function <math>e^{ax}</math>, a constant in the function doesn't alter the differential.