Exponential Function: Complete Lesson with Properties, Derivative and Interactive Graph

The big picture

The exponential function, written $\exp$ or $x \mapsto e^x$, is the most important function in calculus. It has a unique property: it equals its own derivative.

Definition

$\exp : \mathbb{R} \to \mathbb{R}_+^*$

The unique function on $\mathbb{R}$ such that:

• $\exp(0) = 1$
• For all real $x$, $\exp'(x) = \exp(x)$ (derivative equals the function itself)

Also written $\exp(x) = e^x$, where $e \approx 2.71828\ldots$

The 5 essential algebraic properties

These look like power rules — they come up all the time:

Property	Formula
Sum in exponent	$e^{a + b} = e^a \times e^b$
Difference	$e^{a - b} = \dfrac{e^a}{e^b}$
Opposite	$e^{-a} = \dfrac{1}{e^a}$
Power	$(e^a)^n = e^{na}$
Root	$e^{a/2} = \sqrt{e^a}$

Derivative

Fundamental: $(e^x)' = e^x$

General (chain rule): $(e^{u(x)})' = u'(x) \cdot e^{u(x)}$

Move the slider below: at every $x$, the tangent slope equals the function’s value.

x₀ =0.5

Point

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Pente de la tangente (nombre dérivé)

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Examples:

• $(e^{3x})' = 3 e^{3x}$
• $(e^{-x})' = -e^{-x}$
• $(e^{x^2})' = 2x \cdot e^{x^2}$

Monotonicity

For all $x \in \mathbb{R}$, $e^x > 0$ (exponential is always strictly positive).

So $(e^x)' = e^x > 0$ ⇒ $\exp$ is strictly increasing on $\mathbb{R}$.

Consequence: $\exp$ is bijective from $\mathbb{R}$ to $\mathbb{R}_+^*$. Its inverse is the natural log $\ln$.

Limits to memorize

Limit	Value
$\lim_{x \to +\infty} e^x$	$+\infty$
$\lim_{x \to -\infty} e^x$	$0$
$\lim_{x \to +\infty} \dfrac{e^x}{x}$	$+\infty$ (comparative growth)
$\lim_{x \to +\infty} \dfrac{e^x}{x^n}$	$+\infty$ for all $n$
$\lim_{x \to -\infty} x \cdot e^x$	$0$
$\lim_{x \to 0} \dfrac{e^x - 1}{x}$	$1$ (derivative at 0)

Comparative growth: exponential beats any power at $+\infty$.

Equations and inequalities

Since $\exp$ is strictly increasing:

• $e^a = e^b \iff a = b$
• $e^a < e^b \iff a < b$

Example 1: equation

Solve $e^{2x - 3} = e^{x + 1}$.

By bijection: $2x - 3 = x + 1$, so $x = 4$.

Example 2: with a constant

Solve $e^x = 5$.

Using $\ln$: $x = \ln(5) \approx 1.61$.

Example 3: inequality

Solve $e^{2x - 1} \geq e^{3x + 2}$.

By strict growth: $2x - 1 \geq 3x + 2 \iff x \leq -3$.

Full study: $f(x) = x e^{-x}$

A classic exam question.

Domain: $\mathbb{R}$.

Derivative: $f'(x) = e^{-x}(1 - x)$.

Sign of $f'$: since $e^{-x} > 0$, sign of $(1 - x)$:

• $f' > 0$ for $x < 1$
• $f' < 0$ for $x > 1$

Extremum: max at $x = 1$, $f(1) = 1/e \approx 0.37$.

Limits:

• As $x \to -\infty$: $e^{-x} \to +\infty$ and $x \to -\infty$, so $f(x) \to -\infty$.
• As $x \to +\infty$: $x/e^x \to 0$.

Real-world applications

• Population growth: $N(t) = N_0 \cdot e^{kt}$
• Radioactive decay: $N(t) = N_0 \cdot e^{-\lambda t}$
• Capacitor charging: $u(t) = U(1 - e^{-t/\tau})$
• Continuous compound interest: $C(t) = C_0 \cdot e^{rt}$

Common mistakes

1. Confusing $e^{a+b}$ with $e^a + e^b$: it’s a product ($e^a \times e^b$), not a sum.
2. Forgetting $u'$ in chain rule: $(e^{2x})' = 2 e^{2x}$, NOT $e^{2x}$.
3. Thinking $e^{-x}$ can be negative: NO, exponential is always > 0.
4. $\ln$ and $e$ are inverses: $\ln(e^x) = x$ and $e^{\ln(x)} = x$ (for $x > 0$).

Solved exercises

Exercise 1

Simplify $e^{2x} \cdot e^{-3x}$.

Show solution

Sum exponents: $e^{-x}$.

Exercise 2

Solve $e^{x^2} = e^{2x + 3}$.

Show solution

By bijection: $x^2 = 2x + 3$, so $x^2 - 2x - 3 = 0$. $\Delta = 16$, roots $-1$ and $3$.

Exercise 3

Study monotonicity of $g(x) = e^{x^2 - 2x}$.

Show solution

$g'(x) = (2x - 2) e^{x^2 - 2x}$. Since $e^{...} > 0$:

• $g' < 0$ for $x < 1$
• $g' > 0$ for $x > 1$

$g$ decreases on $(-\infty, 1]$, increases on $[1, +\infty)$. Min at $x = 1$: $g(1) = 1/e$.

Takeaways

• $\exp(0) = 1$, $\exp' = \exp$.
• Always strictly positive, strictly increasing, bijective.
• $e^{a+b} = e^a e^b$, $e^{-a} = 1/e^a$.
• Chain rule: $(e^u)' = u' e^u$.
• Beats any power at $+\infty$.
• Inverse: $\ln$.

Going further

• Related tips: Exponential beats any power, Essential derivatives.
• Next: the $\ln$ function (inverse of $\exp$).
• Applications: linear ODE $y' = ay$, solution $y = Ce^{ax}$.