Editing Logarithm (section)

===Derivative and antiderivative===
[[File:Logarithm derivative.svg|thumb|The graph of the [[natural logarithm]] (green) and its tangent at {{math|''x'' {{=}} 1.5}} (black)|alt=A graph of the logarithm function and a line touching it in one point.]]
Analytic properties of functions pass to their inverses.<ref name=LangIII.3 /> Thus, as {{math|1=''f''(''x'') = {{mvar|b}}<sup>''x''</sup>}} is a continuous and [[differentiable function]], so is {{math|log<sub>''b''</sub>&thinsp;''y''}}. Roughly, a continuous function is differentiable if its graph has no sharp "corners". Moreover, as the [[derivative]] of {{math|''f''(''x'')}} evaluates to {{math|ln(''b'') ''b''<sup>''x''</sup>}} by the properties of the [[exponential function]], the [[chain rule]] implies that the derivative of {{math|log<sub>''b''</sub>&thinsp;''x''}} is given by<ref name="LangIV.2"/><ref>{{citation |work=Wolfram Alpha |title=Calculation of ''d/dx(Log(b,x))'' |publisher=[[Wolfram Research]] |access-date=15 March 2011 |url=http://www.wolframalpha.com/input/?i=d/dx(Log(b,x)) }}</ref>
<math display="block">\frac{d}{dx} \log_b x = \frac{1}{x\ln b}. </math>
That is, the [[slope]] of the [[tangent]] touching the graph of the {{math|base-''b''}} logarithm at the point {{math|(''x'', log<sub>''b''</sub>&thinsp;(''x''))}} equals {{math|1/(''x'' ln(''b''))}}.

The derivative of {{Math|ln(''x'')}} is {{Math|1/''x''}}; this implies that {{Math|ln(''x'')}} is the unique [[antiderivative]] of {{math|1/''x''}} that has the value 0 for {{math|1=''x'' = 1}}. It is this very simple formula that motivated to qualify as "natural" the natural logarithm; this is also one of the main reasons of the importance of the [[E (mathematical constant)|constant&nbsp;{{Mvar|e}}]].

The derivative with a generalized functional argument {{math|''f''(''x'')}} is
<math display="block">\frac{d}{dx} \ln f(x) = \frac{f'(x)}{f(x)}.</math>
The quotient at the right hand side is called the [[logarithmic derivative]] of ''{{Mvar|f}}''. Computing {{math|''f<nowiki>'</nowiki>''(''x'')}} by means of the derivative of {{math|ln(''f''(''x''))}} is known as [[logarithmic differentiation]].<ref>{{Citation | last1=Kline | first1=Morris | author1-link=Morris Kline | title=Calculus: an intuitive and physical approach | publisher=[[Dover Publications]] | location=New York | series=Dover books on mathematics | isbn=978-0-486-40453-0 | year=1998}}, p.&nbsp;386</ref> The antiderivative of the [[natural logarithm]] {{math|ln(''x'')}} is:<ref>{{citation |work=Wolfram Alpha |title=Calculation of ''Integrate(ln(x))'' |publisher=Wolfram Research |access-date=15 March 2011 |url=http://www.wolframalpha.com/input/?i=Integrate(ln(x)) }}</ref>
<math display="block">\int \ln(x) \,dx = x \ln(x) - x + C.</math>
[[List of integrals of logarithmic functions|Related formulas]], such as antiderivatives of logarithms to other bases can be derived from this equation using the change of bases.<ref>{{Harvard citations|editor1-last=Abramowitz|editor2-last=Stegun|year=1972 |nb=yes|loc=p. 69}}</ref>