Derivative of Natural Logarithm Function/Proof 3
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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
This proof assumes the definition of the natural logarithm as the inverse of the exponential function as defined by differential equation:
- $y = \dfrac {\d y} {\d x}$
- $y = e^x \iff \ln y = x$
\(\ds \frac {\d y} {\d x}\) | \(=\) | \(\ds y\) | Definition of Exponential Function | |||||||||||
\(\ds \int \frac 1 y \rd y\) | \(=\) | \(\ds \int \rd x\) | Solution to Separable Differential Equation | |||||||||||
\(\ds \) | \(=\) | \(\ds x + C_0\) | Integral of Constant where that constant is $1$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \ln y + C_0\) | Definition 2 of Natural Logarithm: $x = \ln y$ |
The result follows from the definition of the antiderivative and the defined initial condition:
- $\tuple {x_0, y_0} = \tuple {0, 1}$
$\blacksquare$