Derivatives of eˣ and aˣ — proofs — Hinglish notes

4.1.19 · Maths › Calculus I — Limits & Derivatives

1. Woh ek limit jis par sab kuch tika hai

Isse $f (x) = a^{x}$ par apply karo jahan $a > 0$ ho. KAISE: exponent law $a^{x + h} = a^{x} a^{h}$ use karo.

f^{'} (x) = h \to 0 lim \frac{a ^{x + h} - a ^{x}}{h} = h \to 0 lim \frac{a ^{x} a ^{h} - a ^{x}}{h} = a^{x} \cdot isse M (a) kehte hain h \to 0 lim \frac{a ^{h} - 1}{h} .

Toh har exponential ke liye:

\frac{d}{d x} a^{x} = M (a) a^{x}, M (a) = h \to 0 lim \frac{a ^{h} - 1}{h}

Derivative wahi function hai, bas constant $M (a)$ se scale hoke.

2. $e$ ko us base ke roop mein define karna jo $M (a) = 1$ banata hai

Yeh $e \approx 2.71828$ ko define karne ka ek clean tarika hai. Isse:

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

Consistency check (pehle forecast, phir verify): kya yeh limit definition famous $e = lim_{n \to \infty} (1 + \frac{1}{n})^{n}$ se agree karti hai? $h$ ko chhota maano: $e^{h} - 1 \approx h$ ka matlab hai $e^{h} \approx 1 + h$ , yani $e \approx (1 + h)^{1/ h}$ . $h = 1/ n \to 0$ lene par $e = lim (1 + 1/ n)^{n}$ milta hai. ✓ Same number.

3. $M (a)$ exactly nikalna: $M (a) = ln a$

Hum $M (a)$ ko ek mystery limit ki tarah nahi chhodna chahte. KAISE: kisi bhi base ko $e$ ki terms mein likho.

$a = e^{l n a} ⟹ a^{x} = (e^{l n a})^{x} = e^{(l n a) x} .$

Ab $e^{(l n a) x}$ ko chain rule se differentiate karo (outer $e^{u}$ , inner $u = (ln a) x$ ):

\frac{d}{d x} a^{x} = \frac{d}{d x} e^{(l n a) x} = e^{(l n a) x} \cdot (ln a) = a^{x} ln a .

Figure — Derivatives of eˣ and aˣ — proofs

4. Worked examples

5. Common mistakes (steel-manned)

Recall Feynman style: ek 12-saal ke bachche ko samjhao

Socho aisa paisa jo grow karta hai. Ek normal curve jaise $x^{2}$ jab upar jaata hai toh speed up karta hai, lekin uska slope ek alag shape hota hai. $e^{x}$ woh magic curve hai jahan abhi kitna steep hai woh exactly abhi kitna tall hai uske barabar hai. Toh agar yeh $5$ units tall hai, toh speed $5$ se climb kar raha hai. Doosre growth-curves jaise $2^{x}$ ya $3^{x}$ ke liye, climb-speed height times ek fixed "personality number" hai — aur woh number base ka natural log hota hai. $e$ woh ek base hai jiska personality number exactly $1$ hai, isliye mathematicians ise natural kehte hain.

Active recall

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

kya hai?

e^{x}

(yeh apna khud ka derivative hai).

\frac{d}{d x} a^{x}

kya hai?

a^{x} ln a

Difference quotient mein

a^{x}

factor kyun hota hai

a^{x} M (a)

ki tarah?

Kyunki

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

, aur

a^{x}

h

mein constant hai isliye limit se bahar aa jaata hai;

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

Limit

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

kiske barabar hai, aur yeh kya represent karta hai?

ln a

; yeh

a^{x}

ka slope

x = 0

par hai.

Is limit ke through

e

kaise define hota hai?

e

woh base hai jिसके liye

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

ho.

e

\frac{d}{d x} a^{x}

derive karo.

a^{x} = e^{(l n a) x}

; chain rule se

e^{(l n a) x} \cdot ln a = a^{x} ln a

milta hai.

\frac{d}{d x} e^{5 x}

kya hai aur sirf

e^{5 x}

kyun nahi?

5 e^{5 x}

; chain rule inner derivative

5

se multiply karta hai.

2^{x^{2}}

differentiate karo.

2^{x^{2}} (2 x) ln 2

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

" galat kyun hai?

Power rule fixed exponent/varying base ke liye hai; yahan exponent vary karta hai, isliye apply nahi hota.

2^{x}

ka slope

x = 0

par?

ln 2 \approx 0.693

Connections

Chain Rule — $a^{x} = e^{(l n a) x}$ ko derivative mein badalne ke liye zaroori hai.
The number e — definitions — $(1 + 1/ n)^{n}$ vs yahan use ki gayi limit definition.
Natural Logarithm ln x — iska derivative $1/ x$ inverse companion proof hai.
Limits — first principles definition of derivative
Exponential Growth and Decay — applications jahan self-derivative model hai.
Power Rule — contrast: fixed exponent vs fixed base.