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: $9$