Natural exponential function eˣ — graph, derivative preview

3.2.4 · Maths › Exponentials & Logarithms

$e^{x}$ KYA hai?

WHY koi special base matter karti hai? Ek general $a^{x}$ ke liye, derivative hai $\frac{d}{d x} a^{x} = a^{x} \cdot k$ , jahan $k$ koi constant hai jo $a$ par depend karta hai. Hum woh base chahte hain jiske liye $k = 1$ ho, taaki function differentiation ke under perfectly self-replicating ho. Woh base $e$ define ki jaati hai.

HOW karte hain hum key slope fact ko scratch se derive?

Hum shuru karte hain definition of the derivative (first principles) se:

$\frac{d}{d x} a^{x} = lim_{h \to 0} \frac{a ^{x + h} - a ^{x}}{h}$

Yeh step kyun? Derivative by definition average rate of change ka limit hai; hum abhi koi rule assume nahi kar sakte.

$a^{x}$ ko factor out karo ( $h$ par depend nahi karta):

$= a^{x} lim_{h \to 0} \frac{a ^{h} - 1}{h}$

Yeh step kyun? $a^{x + h} = a^{x} \cdot a^{h}$ index law se, aur $a^{x}$ limit variable $h$ ke respect mein constant hai.

Us limit ko $k (a) = h \to 0 lim \frac{a ^{h} - 1}{h}$ kaho. Toh:

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

Yeh kyun matter karta hai: derivative ki shape wahi function wapas hai, bas $k (a)$ se scale hoke.

Numerically limit check karo: $a = 2$ ke liye, $k \approx 0.693$ ; $a = 3$ ke liye, $k \approx 1.099$ . Value $k = 1$ beech mein baithe hai, $a = e \approx 2.718$ par.

Graph — har feature explain kiya gaya

Figure — Natural exponential function eˣ — graph, derivative preview

Feature	Value	WHY
Guzarta hai	$(0, 1)$ se	$e^{0} = 1$ (kuch bhi $^{0} = 1$ )
Guzarta hai	$(1, e) \approx (1, 2.72)$ se	$e$ ki definition
Range	$y > 0$	$e^{x}$ kabhi zero/negative nahi; positive base kisi bhi power par positive rehti hai
Horizontal asymptote	$y = 0$ jab $x \to - \infty$	$e^{- x} = 1/ e^{x} \to 0$
Behaviour jab $x \to + \infty$	$\to + \infty$	bina bound ke badhta hai, kisi bhi polynomial se tez
Slope at $(0, 1)$	$= 1$	defining property
Slope at kisi bhi $(x, e^{x})$	$= e^{x}$ (height ke barabar!)	$\frac{d}{d x} e^{x} = e^{x}$
Concavity	hamesha concave up	$\frac{d ^{2}}{d x ^{2}} e^{x} = e^{x} > 0$

Worked Examples

Common Mistakes (Steel-manned)

Recall Feynman: ek 12-saal ke baache ko explain karo

Socho ek magic snowball pahaad se neeche lud rahi hai. Jitni badi hoti hai, utni tezi se badhti hai — aur uski speed exactly utni hai jitni woh already badi hai. $e^{x}$ wahi math ki snowball hai. Curve par jahan bhi khado, curve kitni oonchi hai woh exactly batata hai kitni steep hai. Aur number $e \approx 2.72$ woh special "growth speed" hai jo yeh perfect matching possible banata hai.

Active Recall Flashcards

What is the derivative of

e^{x}

e^{x}

(yeh apna khud ka derivative hai)

What defines the base

e

among all

a^{x}

Woh base jiske graph ka slope exactly

1

x = 0

par, yaani

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

Approximate value of

e

2.71828 \dots

(irrational)

What point does

y = e^{x}

always pass through, and its slope there?

(0, 1)

aur wahan slope

1

Range of

e^{x}

y > 0

(saare positive reals)

Horizontal asymptote of

e^{x}

y = 0

jab

x \to - \infty

Why isn't

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

Power rule ko variable base aur constant exponent chahiye; yahan variable exponent mein hai

Tangent line to

e^{x}

x = 0

y = x + 1

Slope of

e^{x}

at a general point

(x, e^{x})

e^{x}

— yeh height ke barabar hai

Derivative of

e^{3 x}

3 e^{3 x}

(chain rule)

e^{x}

concave up or down everywhere?

Concave up, kyunki second derivative

e^{x} > 0

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