Derivative of Natural Logarithm of Function

Theorem

Let $u$ be a differentiable real function of $x$.

Let $\ln u$ be the natural logarithm of $u$.

Then:

$\map {\dfrac \d {\d x} } {\ln u} = \dfrac 1 u \dfrac {\d u} {\d x}$

Proof

\(\ds \map {\frac \d {\d x} } {\ln u}\)

\(=\)

\(\ds \map {\frac \d {\d u} } {\ln u} \frac {\d u} {\d x}\)

Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds \dfrac 1 u \frac {\d u} {\d x}\)

Derivative of Natural Logarithm Function

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives of Exponential and Logarithmic Functions: $13.27$
• 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $10$