Derivative of Natural Logarithm Function/Proof 1
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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): $\S 14.1$