Derivative of Exponential Function/Corollary 1

Corollary to Derivative of Exponential Function

Let $\exp x$ be the exponential function.

Let $a \in \R$.

Then:

$\map {\dfrac \d {\d x} } {\map \exp {a x} } = a \map \exp {a x}$


Proof

\(\ds \map {\dfrac \d {\d x} } {\map \exp {a x} }\) \(=\) \(\ds a \map {\dfrac \d {\map \d {a x} } } {\map \exp {a x} }\) Derivative of Function of Constant Multiple
\(\ds \) \(=\) \(\ds a \map \exp {a x}\) Derivative of Exponential Function

$\blacksquare$


Sources

  • 1944: R.P. Gillespie: Integration (2nd ed.) ... (previous) ... (next): Chapter $\text {II}$: Integration of Elementary Functions: $\S 7$. Standard Integrals: $3$.
  • 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $6$: Derivatives
  • 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $7$: Derivatives
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