Natural exponential function eˣ — graph, derivative preview: Foundations

This page assumes nothing. We list every symbol and idea the parent note leans on, in an order where each one only uses things already built. Read top to bottom; by the end the parent note should feel obvious.

1. Powers and exponents — $a^{x}$

The picture: think of a ladder. Each rung multiplies you by $a$ . $a^{3}$ means climb $3$ rungs from the floor: $a^{3} = 1 \times a \times a \times a .$

From whole steps to any exponent — and why $a > 0$

The index laws we will actually use

Why the topic needs these:

$a^{m + n} = a^{m} \cdot a^{n}$ is the engine of the whole derivative derivation. When the parent writes $a^{x + h} = a^{x} \cdot a^{h}$ , that is this law with $m = x, n = h$ .
$a^{- n} = 1/ a^{n}$ explains why (as we'll see) $e^{- x}$ is a small positive number, never negative — a reciprocal of a positive is still positive.
$a^{0} = 1$ explains why every exponential graph passes through the point $(0, 1)$ .

(We deliberately used the plain base $a = 2$ here, because the special base $e$ — the main character of this topic — isn't defined until section 2.)

2. The special number $e$

The picture: on the number line, $e$ sits just past $2.7$ , between $2$ and $3$ — and section 1 tells us $e^{x}$ therefore has a full unbroken graph.

Figure — Natural exponential function eˣ — graph, derivative preview

Figure s01 — a technical-drawing number line with the amber marker for $e \approx 2.718$ ; alongside it, the three ladders $2^{x}, e^{x}, 3^{x}$ compared at $x = 1$ , showing $e$ is the base wedged between $2$ and $3$ that will turn out to be "just right".

Now that $e$ is defined, we can safely reuse the section-1 laws with base $e$ :

$e^{x + h} = e^{x} \cdot e^{h}$ (product law), $e^{- 1} = \frac{1}{e} \approx 0.368$ , $e^{0} = 1$ .

3. Functions and the $f (x)$ notation

The picture: a box with an input arrow ( $x$ ) and an output arrow ( $f (x)$ ).

4. Slope — what "steepness" means as a picture

The picture: a right triangle riding on the line — horizontal leg is the run, vertical leg is the rise.

Figure s02 — a cyan straight line on blueprint grid; an amber right triangle underneath shows the horizontal "run" (change in $x$ ) and vertical "rise" (change in $y$ ) whose ratio is the slope.

A big slope = steep climb.
A slope of $0$ = flat.
A negative slope = going downhill.

Figure s03 — the cyan curve $y = e^{x}$ with an amber tangent line just touching it at one white point; the tangent's slope is the curve's slope exactly there.

5. The derivative and the limit — measuring instantaneous slope

We know slope for a straight line. To get the slope of a curve at a point we sneak up on it.

The picture: a secant line cutting the curve at two points. As we slide the second point closer (shrink $h$ ), the secant swings toward the tangent.

Figure s04 — two grey secant lines through $y = e^{x}$ for a large and a smaller step $h$ ; as $h$ shrinks toward $0$ the secants swing onto the amber tangent line. This is the picture of a limit.

Putting the pieces together — how $e$ is really defined

Feed $f (x) = a^{x}$ into the derivative and use index law $a^{x + h} = a^{x} a^{h}$ (valid since $a > 0$ , and valid for the real step $h$ thanks to section 1): $\frac{d}{d x} a^{x} = lim_{h \to 0} \frac{a ^{x + h} - a ^{x}}{h} = lim_{h \to 0} \frac{a ^{x} a ^{h} - a ^{x}}{h} = a^{x} call this k (a) h \to 0 lim \frac{a ^{h} - 1}{h} .$

What just happened, in words: the derivative of any exponential is the same function back, multiplied by a constant $k (a)$ that depends only on the base. The number $e$ is defined as the base making that constant equal to $1$ : $lim_{h \to 0} \frac{e ^{h} - 1}{h} = 1 ⟹ \frac{d}{d x} e^{x} = e^{x} .$

6. Concave up — the shape word

How these foundations feed the topic

Below is a dependency map: read it top-to-bottom, following the arrows. Each box is one foundation from this page; each arrow means "you need the box at the tail before the box at the head makes sense."

Start top-left at "Powers a^x and index laws" (section 1). It splits two ways: it fixes the special base e (you can't pin down $e$ without powers), and it powers function notation.
Follow the chain function → graph → slope → tangent line: each idea is drawn on the one before it — a graph needs a function, slope needs a graph, a tangent needs slope.
Tangent line feeds the limit (we slide a secant toward the tangent), and both tangent line and limit flow directly into the derivative — the derivative is the tangent's slope computed by a limit. Notice powers also arrows straight into the derivative, because the derivation reuses the index law.
Finally e and derivative meet at "Natural exponential e^x", which delivers the headline "slope equals height", from which concave up follows.

Equipment checklist

What does the exponent in

a^{b}

tell you to do when

b

is a whole number?

Start at

1

and multiply by the base

a

, exactly

b

times.

Why must the base satisfy

a > 0

for

a^{x}

to work for every real

x

Otherwise roots like

a^{1/2} = a

aren't real, and the graph tears into gaps; positivity keeps

a^{h}

real for all small

h

Why does

a^{x}

have a single well-defined value even for irrational

x

Fractions creeping toward

x

give powers that are squeezed into one value (no jumps for

a > 0

), so the limit exists and is unique.

Rewrite

a^{x + h}

using an index law, and note which exponents it's valid for.

a^{x + h} = a^{x} \cdot a^{h}

; valid for any real exponents, including a tiny real step

h

Why is

e^{- x}

always positive?

e^{- x} = 1/ e^{x}

, a positive divided by a positive, so it stays positive (just small).

What is

a^{0}

for any

a > 0

, and which graph point does it fix?

a^{0} = 1

, so every exponential passes through

(0, 1)

Approximate value of

e

, and is it exact?

e \approx 2.71828 \dots

; it is irrational, so no finite decimal is exact.

What does

f (x)

mean?

The output of the function machine

f

when fed the input

x

What do

y

f (x)

and "height of the curve" have in common?

They are three names for the same thing — the output plotted on the vertical axis.

Define slope of a straight line in words.

Rise over run — the change in

y

divided by the change in

x

What is the tangent line at a point on a curve?

The straight line just touching the curve there; its slope is the curve's slope at that point.

What is

lim_{h \to 0} (\dots)

asking?

The value the expression homes in on as

h

gets arbitrarily close to

0

(without being

0

Why may

a^{x}

be pulled outside

lim_{h \to 0}

For a fixed

x

it contains no

h

, so it's a constant with respect to the limit variable and slides out front.

Write the first-principles definition of the derivative.

\frac{d}{d x} f (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}

What single property defines the base

e

The base whose derivative constant is

1

lim_{h \to 0} \frac{e ^{h} - 1}{h} = 1

, so

\frac{d}{d x} e^{x} = e^{x}

What does "concave up" look like?

A smile / valley shape — the slope keeps increasing as you move right.

Connections

Parent: eˣ graph & derivative preview
Euler's number e — definition & limit
General exponential a^x and its derivative
Natural logarithm ln x as inverse of e^x
Chain rule
Taylor series of e^x
Exponential growth and decay models