Derivative of General Logarithm of Function
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Theorem
Let $u$ be a differentiable real function of $x$.
Let $a \in \R_{>0}$ such that $a \ne 1$
Let $\log_a u$ be the logarithm to base $a$ of $u$.
Then:
- $\map {\dfrac \d {\d x} } {\log_a u} = \dfrac {\log_a e} u \dfrac {\d u} {\d x}$
Proof
\(\ds \map {\frac \d {\d x} } {\log_a u}\) | \(=\) | \(\ds \map {\frac \d {\d u} } {\log_a u} \frac {\d u} {\d x}\) | Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds \dfrac {\log_a e} u \frac {\d u} {\d x}\) | Derivative of General Logarithm Function |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives of Exponential and Logarithmic Functions: $13.26$
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $9$