Exponential Function

The exponential function is a unique mathematical tool used to model phenomena that grow or decay at a rate proportional to their size, such as population growth, radioactive decay, or compound interest. Its most remarkable property is that it is its own derivative.

Definition

Definition Exponential Function

There exists a unique differentiable function $f$ on $\mathbb{R}$, called the exponential function and denoted by $\exp$, such that:

$f'(x) = f(x)$ for all $x \in \mathbb{R}$ (the function is equal to its own derivative).
$f(0) = 1$.

Proposition Derivative Property

By definition, for all $x \in \mathbb{R}$:$$\exp'(x) = \exp(x)$$

Algebraic Properties and Notation

Proposition Fundamental Functional Relation

For all real numbers $x$ and $y$:$$\exp(x + y) = \exp(x) \times \exp(y)$$

Proposition Strict Positivity

The function $\exp$ is strictly positive on $\mathbb{R}$: for all $x \in \mathbb{R}$, $\exp(x) > 0$.

Proof

For any real $x$, $\exp(x) = \exp\left(\frac{x}{2} + \frac{x}{2}\right) = \left[\exp\left(\frac{x}{2}\right)\right]^2$.
Since a square is always non-negative, $\exp(x) \geqslant 0$.
We know $\exp(x) \times \exp(-x) = \exp(x-x) = \exp(0) = 1$.
If there existed an $x$ such that $\exp(x) = 0$, then $0 \times \exp(-x) = 1$, which means $0 = 1$. This is impossible.
Therefore, $\exp(x)$ is never zero and must be strictly positive.

Definition The Constant $e$ and Power Notation

The number $\exp(1)$ is denoted by $e$. Its approximate value is $e \approx 2.718$.For any real $x$, it follows that $\exp(x) = \exp(1 \cdot x) = [\exp(1)]^x = e^x$. We now note:$$\exp(x) = e^x$$

Proposition Algebraic Rules

For all real numbers $x, y$ and any integer $n$:

$e^0 = 1$ and $e^1 = e$
$e^{x+y} = e^x \times e^y$
$e^{-x} = \dfrac{1}{e^x}$
$e^{x-y} = \dfrac{e^x}{e^y}$
$(e^x)^n = e^{nx}$

Study of the Function

Proposition Variations

The exponential function is strictly increasing on $\mathbb{R}$.

Proof

The derivative of $f(x) = e^x$ is $f'(x) = e^x$.
As proved previously, $e^x > 0$ for all real $x$. Since the derivative is strictly positive, the function is strictly increasing.

Proposition Derivative of $e^{u(x)}$

Let $u$ be a differentiable function on an interval $I$. The function $f$ defined by $f(x) = e^{u(x)}$ is differentiable on $I$ and its derivative is:$$\textcolor{colorprop}{\left(e^{u(x)}\right)' = u'(x) e^{u(x)}}$$Linear Application:
In the specific case where $u(x) = ax + b$ (with $a$ and $b$ real constants), the derivative is:$$\textcolor{colorprop}{\left(e^{ax+b}\right)' = a e^{ax+b}}$$

Example

Let $f(x)=e^{5x-3}$. Set $u(x)=5x-3$, so $u'(x)=5$. Therefore,$$f'(x)=u'(x)\,e^{u(x)}=5e^{5x-3}.$$

Proposition Growth and Decay Models

Let $k$ be a non-zero real constant. The variation of the function $f: t \mapsto e^{kt}$ depends on the sign of $k$:

If $\boldsymbol{k > 0}$, the function is strictly increasing (Exponential Growth).
If $\boldsymbol{k < 0}$, the function is strictly decreasing (Exponential Decay).