Derivative of Constant to Power of Function

Theorem

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

Let $a \in \R_{>0}$.

Let $a^u$ be $a$ raised to the power of $u$.

Then:

$\map {\dfrac \d {\d x} } {a^u} = a^u \ln a \dfrac {\d u} {\d x}$

Proof

\(\ds \map {\frac \d {\d x} } {a^u}\)

\(=\)

\(\ds \map {\frac \d {\d u} } {a^u} \frac {\d u} {\d x}\)

Chain Rule for Derivatives

\(\ds \)

\(=\)

\(\ds a^u \ln a \frac {\d u} {\d x}\)

Derivative of Power of Constant

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives of Exponential and Logarithmic Functions: $13.28$
• 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $11$