Derivative of Exponential of Function
Jump to navigation
Jump to search
Theorem
Let $u$ be a differentiable real function of $x$.
Let $e^u$ be the exponential function of $u$.
Then:
- $\map {\dfrac \d {\d x} } {e^u} = e^u \dfrac {\d u} {\d x}$
Proof
\(\ds \map {\frac \d {\d x} } {e^u}\) | \(=\) | \(\ds \map {\frac \d {\d u} } {e^u} \frac {\d u} {\d x}\) | Chain Rule for Derivatives | |||||||||||
\(\ds \) | \(=\) | \(\ds e^u \frac {\d u} {\d x}\) | Derivative of Exponential Function |
$\blacksquare$
Sources
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 13$: Derivatives of Exponential and Logarithmic Functions: $13.29$
- 1974: Murray R. Spiegel: Theory and Problems of Advanced Calculus (SI ed.) ... (previous) ... (next): Chapter $4$. Derivatives: Derivatives of Special Functions: $12$