Derivative of Natural Logarithm Function/Proof 2
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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, where the exponential function is defined as the limit of a sequence:
- $e^x := \ds \lim_{n \mathop \to +\infty} \paren {1 + \frac x n}^n$
It also assumes the Laws of Logarithms.
\(\ds \map {\frac \d {\d x} } {\ln x}\) | \(=\) | \(\ds \lim_{\Delta x \mathop \to 0} \frac {\map \ln {x + \Delta x} - \ln x} {\Delta x}\) | Definition of Derivative | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\Delta x \mathop \to 0} \frac {\map \ln {\frac {x + \Delta x} x} } {\Delta x}\) | Difference of Logarithms | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{\Delta x \mathop \to 0} \paren {\frac 1 {\Delta x} \centerdot \map \ln {1 + \frac {\Delta x} x} }\) | ||||||||||||
\(\text {(1)}: \quad\) | \(\ds \) | \(=\) | \(\ds \lim_{\Delta x \mathop \to 0} \paren {\map \ln {\paren {1 + \frac {\Delta x} x}^{1 / \Delta x} } }\) | Natural Logarithm of Power |
Define $u$ as:
\(\ds u\) | \(=\) | \(\ds \dfrac {\Delta x} x\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \Delta x\) | \(=\) | \(\ds u x\) | |||||||||||
\(\ds \leadsto \ \ \) | \(\ds \frac 1 {\Delta x}\) | \(=\) | \(\ds \frac 1 x \cdot \frac 1 u\) |
Hence:
\(\ds \) | \(=\) | \(\ds \lim_{u \mathop \to 0} \paren {\map \ln {\paren {1 + u}^{\frac 1 u \cdot \frac 1 x} } }\) | substituting $u x$ for $\Delta x$ in $(1)$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \lim_{u \mathop \to 0} \paren {\frac 1 x \cdot \map \ln {1 + u}^{\frac 1 u} }\) | Natural Logarithm of Power | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 x \cdot \lim_{u \mathop \to 0} \paren {\map \ln {1 + u}^{\frac 1 u} }\) | factoring out constants | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 x \cdot \lim_{v \mathop \to +\infty} \paren {\map \ln {1 + \frac 1 v}^v}\) | substituting $\dfrac 1 v$ for $u$ | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 x \cdot \ln e^1\) | Limit of Composite Function, Limit definition of $e^x$, Real Natural Logarithm Function is Continuous | |||||||||||
\(\ds \) | \(=\) | \(\ds \frac 1 x\) | Exponential of Natural Logarithm |
$\blacksquare$
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {II}$. Calculus: Differentiation: Standard Differential Coefficients
- For a video presentation of the contents of this page, visit the Khan Academy.