Derivative of Natural Logarithm Function/Proof 1

Theorem

Let $\ln x$ be the natural logarithm function.

Then:

$\map {\dfrac \d {\d x} } {\ln x} = \dfrac 1 x$


Proof

\(\ds \ln x\) \(:=\) \(\ds \int_1^x \dfrac 1 t \rd t\) Definition 1 of Natural Logarithm
\(\ds \frac \d {\d x} \ln x\) \(=\) \(\ds \frac \d {\d x} \int_1^x \dfrac 1 t \rd t\)
\(\ds \) \(=\) \(\ds \frac 1 x\) Fundamental Theorem of Calculus

$\blacksquare$


Sources

  • 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 14.1$
This article is issued from ProofWiki. The text is available under Creative Commons Attribution-ShareAlike License unless otherwise noted. Additional terms may apply for the media files.