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): $\S 14.1$