Derivative of General Logarithm of Function

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: $9$
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