# Derivative of Exponential Function

## Contents

## Theorem

Let $\exp$ be the exponential function.

Then:

- $\map {D_x} {\exp x} = \exp x$

### Corollary 1

Let $c \in \R$.

Then:

- $D_x \left({\exp \left({c x}\right)}\right) = c \exp \left({c x}\right)$

### Corollary 2

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

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

Then:

- $\map {D_x} {a^x} = a^x \ln a$

## Proof 1

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

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

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

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

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

$\blacksquare$

## Proof 2

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

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

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

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

\(\displaystyle \) | \(=\) | \(\displaystyle y\) | |||||||||||

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

$\blacksquare$

## Proof 3

\(\displaystyle \map {D_x} {\ln e^x}\) | \(=\) | \(\displaystyle \map {D_x} x\) | Exponential of Natural Logarithm | ||||||||||

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

\(\displaystyle \leadsto \ \ \) | \(\displaystyle \map {D_x} {e^x}\) | \(=\) | \(\displaystyle e^x\) | multiply both sides by $e^x$ |

$\blacksquare$

## Proof 4

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

That is, let:

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

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:

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

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

\(\displaystyle \) | \(=\) | \(\displaystyle \sum_{k \mathop = 1}^\infty D_x \frac {x^{k - 1} } {\left({k - 1}\right)!}\) | |||||||||||

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

\(\displaystyle \) | \(=\) | \(\displaystyle \exp x\) |

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, $I$ is compact.

From Chain Rule for Derivatives:

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

### Lemma

- $\forall x \in \R : n \ge \left\lceil{\left\vert{x}\right\vert}\right\rceil \implies \left\langle{\dfrac n {n + x} \left({1 + \dfrac x n}\right)^n}\right\rangle$ is increasing.

$\Box$

From the lemma:

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

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

Therefore, for $x \in I$:

\(\displaystyle D_x \exp x\) | \(=\) | \(\displaystyle D_x \lim_{n \mathop \to \infty} \map {f_n} x\) | |||||||||||

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

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

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

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

\(\displaystyle \) | \(=\) | \(\displaystyle \exp x\) |

In particular:

- $D_x \exp x_0 = \exp x_0$

$\blacksquare$

## Also see

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

## Sources

- 1968: Murray R. Spiegel:
*Mathematical Handbook of Formulas and Tables*... (previous) ... (next): $\S 13$: Derivatives of Exponential and Logarithmic Functions: $13.29$ - 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $2 \cdotp 718 \, 281 \, 828 \, 459 \, 045 \, 235 \, 360 \, 287 \, 471 \, 352 \, 662 \, 497 \, 757 \, 247 \, 093 \, 699 \ldots$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $2 \cdotp 71828 \, 18284 \, 59045 \, 23536 \, 02874 \, 71352 \, 66249 \, 77572 \, 47093 \, 69995 \ \ldots$

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