# Derivative of Exponential Function

## Theorem

Let $\exp$ be the exponential function.

Then:

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

### Corollary 1

Let $a \in \R$.

Then:

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

### Corollary 2

Let $a \in \R: a > 0$.

Let $a^x$ be $a$ to the power of $x$.

Then:

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

### Corollary 3

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

## Proof 1

\(\ds \map {\frac \d {\d x} } {\exp x}\) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\map \exp {x + h} - \exp x} h\) | Definition of Derivative | |||||||||||

\(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\exp x \cdot \exp h - \exp x} h\) | Exponential of Sum | |||||||||||

\(\ds \) | \(=\) | \(\ds \lim_{h \mathop \to 0} \frac {\exp x \paren {\exp h - 1} } h\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \exp x \paren {\lim_{h \mathop \to 0} \frac {\exp h - 1} h}\) | Multiple Rule for Limits of Real Functions, as $\exp x$ is constant | |||||||||||

\(\ds \) | \(=\) | \(\ds \exp x\) | Derivative of Exponential at Zero |

$\blacksquare$

## Proof 2

We use the fact that the exponential function is the inverse of the natural logarithm function:

- $y = e^x \iff x = \ln y$

\(\ds \dfrac {\d x} {\d y}\) | \(=\) | \(\ds \dfrac 1 y\) | Derivative of Natural Logarithm Function | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \dfrac {\d y} {\d x}\) | \(=\) | \(\ds \dfrac 1 {1 / y}\) | Derivative of Inverse Function | ||||||||||

\(\ds \) | \(=\) | \(\ds y\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds e^x\) |

$\blacksquare$

## Proof 3

\(\ds \map {\frac \d {\d x} } {\ln e^x}\) | \(=\) | \(\ds \map {\frac \d {\d x} } x\) | Exponential of Natural Logarithm | |||||||||||

\(\ds \leadsto \ \ \) | \(\ds \frac 1 {e^x} \map {\frac \d {\d x} } {e^x}\) | \(=\) | \(\ds 1\) | Chain Rule for Derivatives, Derivative of Natural Logarithm Function, Derivative of Identity Function | ||||||||||

\(\ds \leadsto \ \ \) | \(\ds \map {\frac \d {\d x} } {e^x}\) | \(=\) | \(\ds e^x\) | multiply both sides by $e^x$ |

$\blacksquare$

## Proof 4

This proof assumes the power series definition of $\exp$.

That is, let:

- $\ds \exp x = \sum_{k \mathop = 0}^\infty \frac {x^k} {k!}$

From Series of Power over Factorial Converges, the interval of convergence of $\exp$ is the entirety of $\R$.

So we may apply Differentiation of Power Series to $\exp$ for all $x \in \R$.

Thus we have:

\(\ds \frac \d {\d x} \exp x\) | \(=\) | \(\ds \frac \d {\d x} \sum_{k \mathop = 0}^\infty \frac {x^k} {k!}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \frac k {k!} x^{k - 1}\) | Differentiation of Power Series, with $n = 1$ | |||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 1}^\infty \frac {x^{k - 1} } {\paren {k - 1}!}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \sum_{k \mathop = 0}^\infty \frac {x^k} {k!}\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \exp x\) |

Hence the result.

$\blacksquare$

## Proof 5

This proof assumes the limit definition of $\exp$.

So let:

- $\forall n \in \N: \forall x \in \R: \map {f_n} x = \paren {1 + \dfrac x n}^n$

Let $x_0 \in \R$.

Consider $I := \closedint {x_0 - 1} {x_0 + 1}$.

Let:

- $N = \ceiling {\max \set {\size {x_0 - 1}, \size {x_0 + 1} } }$

where $\ceiling {\, \cdot \,}$ denotes the ceiling function.

From Closed Real Interval is Compact in Metric Space, $I$ is compact.

From Chain Rule for Derivatives:

- $\dfrac \d {\d x} \map {f_n} x = \dfrac n {n + x} \map {f_n} x$

### Lemma

- $\forall x \in \R : n \ge \ceiling {\size x} \implies \sequence {\dfrac n {n + x} \paren {1 + \dfrac x n}^n}$ is increasing.

$\Box$

From the lemma:

- $\forall x \in I: \sequence {\dfrac \d {\d x} \map {f_{n + N} } x}$ is increasing

Hence, from Dini's Theorem, $\sequence {\dfrac \d {\d x} f_{n + N} }$ is uniformly convergent on $I$.

Therefore, for $x \in I$:

\(\ds \frac \d {\d x} \exp x\) | \(=\) | \(\ds \frac \d {\d x} \lim_{n \mathop \to \infty} \map {f_n} x\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds \frac \d {\d x} \lim_{n \mathop \to \infty} \map {f_{n + N} } x\) | Tail of Convergent Sequence | |||||||||||

\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \frac \d {\d x} \map {f_{n + N} } x\) | Derivative of Uniformly Convergent Sequence of Differentiable Functions | |||||||||||

\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \frac n {n + x} \map {f_n} x\) | from above | |||||||||||

\(\ds \) | \(=\) | \(\ds \lim_{n \mathop \to \infty} \map {f_n} x\) | Combination Theorem for Sequences | |||||||||||

\(\ds \) | \(=\) | \(\ds \exp x\) |

In particular:

- $\dfrac \d {\d x} \exp x_0 = \exp x_0$

$\blacksquare$

## Also see

- Equivalence of Definitions of Exponential Function where it is shown that $\dfrac \d {\d x} \exp x = \exp x$ can be used to
*define*the (real) exponential function.

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$ - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Appendix $2$: Table of derivatives and integrals of common functions - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$ - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Appendix: Table $1$: Derivatives

- 2005: Roland E. Larson, Robert P. Hostetler and Bruce H. Edwards:
*Calculus*(8th ed.): $\S 5.4$