Derivative of Natural Logarithm of a x
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Theorem
Let $\ln x$ be the natural logarithm function.
Then:
- $\map {\dfrac \d {\d x} } {\ln a x} = \dfrac 1 x$
Proof
| \(\ds \map {\dfrac \d {\d x} } {\ln a x}\) | \(=\) | \(\ds a \map {\dfrac \d {\d \paren {a x} } } {\ln a x}\) | Derivative of Function of Constant Multiple | |||||||||||
| \(\ds \) | \(=\) | \(\ds a \dfrac 1 {a x}\) | Derivative of Natural Logarithm | |||||||||||
| \(\ds \) | \(=\) | \(\ds \dfrac 1 x\) |