Natural exponential function eˣ — graph, derivative preview: Question bank

Before you start, recall the one fact everything here orbits: for $e^{x}$ , the slope at any point equals the height at that point, because $\frac{d}{d x} e^{x} = e^{x}$ .

True or false — justify

TF1.

e^{x}

is always positive, even when

x

is a large negative number.

True.

e^{- x} = \frac{1}{e ^{x}}

is positive-over-positive, so it stays above zero and only approaches the line

y = 0

without ever touching it.

TF2. The derivative of

e^{x}

x e^{x - 1}

False. That is the power rule, which needs the variable in the base. Here the variable sits in the exponent, so the correct derivative is

e^{x}

itself.

TF3.

e

is exactly equal to

2.718

False.

e = 2.71828 \dots

is irrational; it is defined by the slope-1 property, and no finite decimal captures it.

TF4. The slope of

y = e^{x}

can be negative somewhere.

False. The slope equals the height

e^{x}

, which is always positive, so the curve is always increasing — it never has a negative slope anywhere.

TF5.

y = e^{x}

has a maximum or minimum turning point.

False. A turning point needs slope

= 0

, but

e^{x}

is never zero, so the slope

e^{x}

is never zero — no turning points ever.

TF6. Both

e^{x}

and

2^{x}

pass through the point

(0, 1)

True. Any base to the power

0

equals

1

, so every exponential

a^{x}

passes through

(0, 1)

— what makes

e^{x}

special is its slope of exactly

1

there.

TF7.

e^{x}

is concave up for all

x

True. The second derivative is

e^{x}

again, which is always positive, so the curve always bends upward — it never has an inflection point.

TF8. As

x \to + \infty

e^{x}

eventually grows slower than

x^{100}

False.

e^{x}

eventually beats every polynomial, no matter how high the power, because repeated differentiation shrinks a polynomial to zero but leaves

e^{x}

unchanged.

TF9. The tangent to

y = e^{x}

x = 0

is the line

y = x + 1

True. The point is

(0, 1)

and the slope is

e^{0} = 1

, so point-slope gives

y = x + 1

— this is why

e^{x} \approx 1 + x

near zero.

Spot the error

SE1. "

\frac{d}{d x} e^{3 x} = e^{3 x}

Missing the chain rule. The inner function

3 x

has derivative

3

, which multiplies in, giving

\frac{d}{d x} e^{3 x} = 3 e^{3 x}

SE2. "Since

e^{0} = 0

, the graph starts at the origin."

e^{0} = 1

, not

0

— anything to the power zero is one. The graph passes through

(0, 1)

, one unit above the origin.

SE3. "

e^{x} = 0

when

x = - \infty

, so the graph touches the

x

-axis."

y = 0

is an asymptote, approached but never reached. There is no finite

x

(and

- \infty

is not a value) at which

e^{x}

actually equals zero.

SE4. "For

y = a^{x}

the derivative is

a^{x}

for every base

a

Only for

a = e

. In general

\frac{d}{d x} a^{x} = k (a) a^{x}

where

k (a) = lim_{h \to 0} \frac{a ^{h} - 1}{h}

; only when

a = e

does

k = 1

and the extra factor vanish.

SE5. "The slope of

e^{x}

x = 2

2

The slope at

x = 2

equals the height there,

e^{2} \approx 7.39

, not the

x

-value. Confusing the input

x

with the slope is the classic slip.

SE6. "

e^{x} \cdot e^{x} = e^{x^{2}}

Index laws add exponents when bases match:

e^{x} \cdot e^{x} = e^{x + x} = e^{2 x}

. You only get

e^{x^{2}}

from

(e^{x})^{x}

, a different expression.

SE7. "Because

e^{x}

grows fast, its graph must be concave down as it flattens near the asymptote."

It never flattens and bends the same way —

e^{x}

is concave up everywhere, including where it hugs the asymptote on the left; the curve is a gentle upward bowl throughout.

Why questions

WY1. Why is

e

chosen as the "natural" base rather than

2

10

Because

e

is the only base whose exponential is its own derivative with no extra constant, making

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

and every calculus formula clean.

WY2. Why does "slope equals height" force the curve to be always increasing and always concave up?

The height

e^{x}

is positive, so the slope (equal to it) is positive → increasing; and the slope keeps growing as the height grows → concave up.

WY3. Why does

e^{x} \approx 1 + x

work only for small

x

1 + x

is the tangent line at

x = 0

; a straight line only hugs a curved graph near the point of tangency, and

e^{x}

curves away from it as

∣ x ∣

grows.

WY4. Why can't the power rule ever apply to

e^{x}

The power rule differentiates a variable base raised to a constant power (

x^{n}

). In

e^{x}

the base

e

is constant and the variable is the exponent — the opposite structure.

WY5. Why does defining

e

through the limit

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

pin down a single number?

The limit

k (a)

increases smoothly as

a

increases (e.g.

k (2) \approx 0.693

k (3) \approx 1.099

), so exactly one base sits where

k = 1

— that base is

e \approx 2.718

WY6. Why is

e^{- x}

small but never zero for large positive

x

e^{- x} = \frac{1}{e ^{x}}

; as

e^{x}

grows huge, its reciprocal shrinks toward zero, but a reciprocal of a finite positive number is always a positive number, never exactly zero.

Edge cases

EC1. What is the slope of

y = e^{x}

exactly at the

y

-intercept?

It is

1

, since the slope equals the height

e^{0} = 1

there — this slope-1 crossing is the defining feature that separates

e

from every other base.

EC2. What happens to the slope of

e^{x}

x \to - \infty

The slope

\to 0

(from above), because slope equals height and

e^{x} \to 0^{+}

; the curve becomes ever more nearly horizontal but never actually flat.

EC3. Is there any

x

where the graph of

e^{x}

crosses below its own tangent line

y = x + 1

No. Since

e^{x}

is concave up everywhere, it lies on or above every one of its tangent lines, with equality only at the single point of tangency.

EC4. What is

e^{0}

, and why does that value matter for the whole graph?

e^{0} = 1

; it fixes the anchor point

(0, 1)

through which the curve passes, and it is where the defining slope-1 behaviour is measured.

EC5. Does

e^{x}

ever equal a negative number for any real

x

Never. A positive base raised to any real power stays positive, so the entire graph lives strictly above the

x

-axis for all real

x

Connections

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