Derivatives of eˣ and aˣ — proofs: Visual walkthrough

4.1.19 · D2 · Maths › Calculus I — Limits & Derivatives › Derivatives of eˣ and aˣ — proofs

Shuru karne se pehle, teen plain-word anchors jo hume chahiye honge:

Step 1 — "Slope" ka matlab kya hai: rise over run

KYA. $a^{x}$ ki steepness ko point $x$ par measure karne ke liye, hum ek doosra point lete hain jo $h$ step right par hai, $x + h$ par. Hum dono points se guzarne wali straight line draw karte hain aur uska tilt measure karte hain.

KYUN. Ek curve ka koi ek "tilt" nahi hota jaise ek straight line ka hota hai — woh bend karta hai. Toh hum ek trick karte hain: do nearby points chunte hain, unhe join karne wali straight line ka tilt measure karte hain, phir second point ko first ki taraf slide karte hain. Yahi ek curved cheez ki slope ke baare mein honestly baat karne ka tariqa hai.

PICTURE. Figure mein, $x$ par height lower dot hai; $x + h$ par height upper dot hai. Run (magenta) horizontal gap $h$ hai. Rise (orange) vertical gap $a^{x + h} - a^{x}$ hai. Unka ratio tilt hai.

Figure — Derivatives of eˣ and aˣ — proofs

slope of the joining line = tilt of the straight line \frac{a ^{x + h} - a ^{x} rise (orange)}{run (magenta) h}

Jaise-jaise $h$ $0$ ki taraf shrink hota hai, do dots merge ho jaate hain aur joining line true tangent ban jaati hai — exact slope. Woh shrinking $lim_{h \to 0}$ likhi jaati hai, matlab "jis value par yeh ratio home karta hai jab $h$ tiny hota jaata hai." Yeh first-principles definition hai:

f^{'} (x) = h \to 0 lim \frac{f ( x + h ) - f ( x )}{h} .

Yahan $f^{'} (x)$ (padho " $f$ -prime") sirf slope function ka ek naam hai.

Step 2 — Magic split: shape khud ko repeat karta hai

KYA. Hum $f (x) = a^{x}$ ko us ratio mein daalnge aur ek exponent law use karenge: $a^{x + h} = a^{x} \cdot a^{h}$ .

KYUN. Exponent mein add karna = powers ko multiply karna. Yeh law poore proof ka engine hai: yeh hume $x$ -part ko $h$ -part se alag karne deta hai.

PICTURE. Figure dekho: tall bar $a^{x}$ hai, taller bar $a^{x + h}$ hai. Doosra bar sirf pehla bar hai jo factor $a^{h}$ se stretch hua hai — same shape, scaled. Woh scaling factor hi saara $h$ -information carry karta hai.

\frac{a ^{x + h} - a ^{x}}{h} = \frac{a ^{x} a ^{h} stretched bar - a ^{x} original bar}{h} = a^{x} \cdot depends on h only \frac{a ^{h} - 1}{h}

Humne $a^{x}$ ko top ke dono terms se factor out kiya. Jo andar bacha, $\frac{a ^{h} - 1}{h}$ , usme bilkul bhi $x$ nahi — sirf base $a$ aur shrinking step $h$ .

Step 3 — Woh constant jo bachta hai: $M (a)$

KYA. Ab $h \to 0$ karo. Saamne wala factor $a^{x}$ mein $h$ nahi hai, toh woh wahan baitha rehta hai. Bachne wala limit ek single number ban jaata hai jise hum $M (a)$ naam dete hain.

KYUN. Kyunki $a^{x}$ $h$ ki nazar se constant hai (sirf $h$ move kar raha hai), woh seedha limit se bahar nikal jaata hai. Bachne wala limit jo bhi ho, woh base $a$ par depend karta hai lekin iss par nahi ki tum kahaan ho ( $x$ ). Ek number, har base ke liye fixed.

Intuition Kyun bachne wala limit actually exist karta hai (sirf ek hopeful naam nahi)

$h = 0$ par fraction $\frac{a ^{h} - 1}{h}$ padhta hai $\frac{a ^{0} - 1}{0} = \frac{0}{0}$ — point par undefined. Lekin hum kabhi $h = 0$ plug nahi karte; hum dekhte hain ki ratio kya approach karta hai jab $h$ , $0$ ke paas aata hai. Geometrically woh ratio exactly $x = 0$ ke paas curve $a^{x}$ par joining line ka tilt hai (Step 1 ki picture). Jaise-jaise $h$ shrink hota hai, woh joining lines ek limiting tangent line par settle ho jaati hain — left se aur right se agree karte hain, kyunki $a^{x}$ ek smooth, unbroken, steadily-increasing (for $a > 1$ ) ya steadily-decreasing (for $a < 1$ ) curve hai jisme $x = 0$ par koi corner nahi hai. Ek single settled tilt = limit exist karta hai. Toh $M (a)$ har $a > 0$ ke liye ek genuine number hai, koi wish nahi. ( $a = 1$ ke liye curve flat line $1^{x} = 1$ hai, tilt $0$ — limit clearly $0$ hai.)

PICTURE. Figure $x = 0$ ke paas zoom in karta hai. Wahan height $a^{0} = 1$ hai, aur tangent ki steepness wahan exactly $M (a)$ hai — slope-at-the-start. Teen curves ( $2^{x}$ violet, $e^{x}$ magenta, $3^{x}$ orange) point $(0, 1)$ se teen alag tilts par nikalte hain.

f^{'} (x) = a^{x} \cdot M (a) = slope at x = 0 h \to 0 lim \frac{a ^{h} - 1}{h} ⟹ \frac{d}{d x} a^{x} = M (a) a^{x}

Box ko words mein padho: ek exponential ka slope uski apni height hai ek personality number $M (a)$ se multiply karke. Poora baaki page yahi hai: $M (a)$ find karo.

Step 4 — Woh base chunna jo $M = 1$ banata hai: yahi $e$ define karta hai

KYA. Alag-alag bases alag-alag $M (a)$ dete hain (Step 3 ke teen tilts). Hum dhundte hain woh ek base jiska starting slope exactly $1$ ho. Us base ko hum $e$ naam dete hain.

KYUN. Agar $M = 1$ , toh Step 3 ka box collapse ho jaata hai $f^{'} (x) = 1 \cdot a^{x} = a^{x}$ mein — curve apna khud ka slope ban jaata hai, koi leftover factor nahi. Yeh possible sabse cleanest derivative hai, isliye yeh base "natural" ka title earn karta hai.

PICTURE. Figure line $y = x + 1$ overlay karta hai (slope $1$ ka tangent jo $(0, 1)$ se guzarta hai). Curve $2^{x}$ us line ke tilt se below start karta hai (bahut gentle, $M < 1$ ); $3^{x}$ usse above start karta hai (bahut steep, $M > 1$ ). Beech mein kahin woh perfect curve baitha hai jo line ko kiss karta hai — $(0, 1)$ se usse tangent hokar nikalti hai. Woh curve $e^{x}$ hai.

Intuition Kyun exactly ek aisa base

$e$ hai (woh missing "why") Do facts isse pin down karte hain. (1) $M (a)$ steadily $a$ ke saath badhta hai. Ek bada base $a^{h}$ ko small $h$ ke liye faster grow karta hai, isliye start-tilt $M (a)$ sirf increase hi karta hai jab $a$ badhta hai — woh kabhi wapas nahi aata. (Yeh Step 3 ki picture mein dikh sakta hai: $2^{x}$ , $3^{x}$ se gentler hai $(0, 1)$ par, hamesha.) (2) $M (a)$ continuously sweep karta hai $1$ se neeche se $1$ se upar tak: $a = 2$ par tilt $1$ se neeche hai (gentle), $a = 3$ par woh $1$ se upar hai (steep), aur beech mein smoothly move karta hai koi jump nahi. Ek quantity jo steadily aur continuously " $1$ se neeche" se " $1$ se upar" tak badhti hai, woh $1$ ki value ek baar aur sirf ek baar hit karegi — yeh intermediate-value idea hai. Woh single crossing point hi woh base hai jise hum $e$ naam dete hain.

$M (e) = 1$ ko Step 3 ke box mein daalnge:

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

Step 5 — Har base ke liye $M (a)$ pin down karna: yeh $ln a$ hai

KYA. $M (a)$ ko ek mystery limit ki tarah chhod dena unsatisfying hai. Hum kisi bhi base $a > 0$ ko $e$ use karke rewrite karte hain aur Chain Rule se differentiate karte hain.

KYUN. Hum $e^{(stuff)}$ ko pehle se fully samajhte hain. Toh agar hum $a^{x}$ ko $e^{(something) x}$ ki tarah dress up kar sakein, hum clean $e$ -result reuse kar sakte hain. Woh tool jo hume " $e$ of an inner function" differentiate karne deta hai woh chain rule hai: outer slope times inner slope.

PICTURE. Figure rewrite ko horizontal axis ki re-labelling ke roop mein dikhata hai. Har base $a$ $e$ hai jo $ln a$ power se raise kiya gaya hai (yahi $\ln a$ ka matlab hai: woh exponent jo $e$ par daalo toh $a$ mile). Toh $a^{x}$ actually $e^{x}$ hi hai jo $ln a$ factor se faster ya slower chalta hai.

Step by step:

a = e^{l n a} ⟹ a^{x} = (e^{l n a})^{x} = e^{(ln a) inner slope x}

\frac{d}{d x} e^{(l n a) x} = outer slope = itself e^{(l n a) x} \cdot inner slope (ln a) = a^{x} ln a

Step 6 — Har case: steep, gentle, flat, aur falling

KYA. Hum formula $M (a) = ln a$ ko har tarah ke bases $a > 0$ ke against check karte hain, including degenerate ones. Woh rule jise tum apne edges par test nahi kar sakte, woh ek aisa rule hai jis par tumhe trust nahi karna chahiye.

KYUN. Reader ko koi aisa base nahi milna chahiye jisko humne show nahi kiya. Toh: $e$ se bada base, $e$ ke barabar base, $1$ aur $e$ ke beech base, $1$ ke barabar base, aur $1$ se neeche base (decay).

PICTURE. Figure personality number $M (a) = ln a$ ko base $a$ ke against plot karta hai. Dekho curve kahan key values cross karta hai.

Base $a$	$ln a$ = start-slope $M (a)$	Curve kya karta hai
$a > e$ (e.g. $3$ )	$ln 3 \approx 1.10 > 1$	start mein $e^{x}$ se steeper
$a = e$	$ln e = 1$	self-derivative — slope $=$ height
$1 < a < e$ (e.g. $2$ )	$ln 2 \approx 0.69 < 1$	start mein $e^{x}$ se gentler
$a = 1$	$ln 1 = 0$	flat line $1^{x} = 1$ ; slope $0$ ✓
$0 < a < 1$ (e.g. $\frac{1}{2}$ )	$ln \frac{1}{2} \approx - 0.69 < 0$	decay — slope negative, curve girta hai

Ek-picture summary

Upar sab kuch, compressed: first principles se shuru karo, exponent law se split karo, personality number $M (a)$ isolate karo, woh base chunno jo ise $1$ banata hai (woh $e$ hai), aur baaki saare bases ko $e$ ke through rewrite karo taaki $M (a) = ln a$ discover ho.

Step 1 h \to 0 lim \frac{a ^{x + h} - a ^{x}}{h} = Steps 2-3 a^{x} h \to 0 lim \frac{a ^{h} - 1}{h} = M (e) = 1 \Rightarrow e^{x} is its own slope a^{x} M (a) = Steps 4-5 a^{x} ln a

Recall Feynman retelling — plain words mein poora walk

Sirf positive bases $a > 0$ ek unbroken curve banate hain jise slope kiya ja sake, isliye hum unhi ko allow karte hain. Yeh jaanne ke liye ki $a^{x}$ kitna steep hai, main ek tiny step $h$ right par jump karta hoon aur rise over run measure karta hoon. Right jump karna sirf height ko $a^{h}$ se multiply karta hai (exponent mein add karna = multiply karna), isliye $x$ par height seedha calculation se bahar nikal jaati hai. Jo bachta hai woh ek number per base hai — tilt bilkul start par, $x = 0$ par, jahan height $1$ hai. Kyunki curve smooth hai aur wahan koi corner nahi, woh joining lines ek tangent par settle ho jaati hain, toh woh number actually exist karta hai; ise base ki "personality" kaho. Jaise main base badhata hoon, personality steadily aur smoothly $1$ se neeche se $1$ se upar tak badhti hai, toh ise exactly ek baar $1$ se guzarna hi hoga — woh base $e \approx 2.718$ hai, aur $e^{x}$ differentiate karne par khud ko copy karta hai. Finally, kyunki koi bhi base $a$ sirf $e$ raise to $ln a$ hai (aur ek power ki power exponents multiply karte hain), har exponential secretly $e^{x}$ hi hai jo factor $ln a$ se speed up hua — isliye uska personality number $ln a$ hai. Bada base $\to$ bada positive $ln$ (steep climb); base $1 \to ln 1 = 0$ (flat line); $1$ se neeche base $\to$ negative $ln$ (woh girta hai). Ek tidy function, $ln a$ , har case sahi karta hai.

Active recall

Base

a

strictly positive kyun hona chahiye?

Agar

a \leq 0

, toh

a^{1/2} = a

jaisi values real nahi hain, isliye

a^{x}

ek unbroken curve nahi hai; sirf

a > 0

hi har real

x

ke liye ek real value deta hai.

Limit

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

actually exist kyun karta hai?

Ratio

x = 0

ke paas joining lines ka tilt hai; smooth exponential par yeh dono sides se ek tangent par settle ho jaate hain, isliye limit ek genuine number hai.

Exactly ek base

e

kyun hai jiske liye

M (e) = 1

M (a) = ln a

steadily aur continuously

1

se neeche se

1

se upar badhta hai jab

a

badhta hai, isliye intermediate-value idea se woh exactly ek base par

1

ke barabar hota hai.

M (a)

geometrically kya represent karta hai?

a^{x}

ka slope

x = 0

par, jahan height

a^{0} = 1

hai.

Kaun sa exponent rule

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

e^{(l n a) x}

mein turn karta hai?

Ek power ki power exponents multiply karta hai:

(b^{c})^{d} = b^{c d}

M (a)

finally kiske barabar hai?

ln a

a = 1

par

M (a)

kya hai, aur curve kaisa dikhta hai?

ln 1 = 0

;

1^{x} = 1

ek flat line hai jiska slope

0

hai.

0 < a < 1

ke liye

M (a)

ka kya sign hai, aur iska kya matlab hai?

Negative; curve decay karta hai (girta hai).

Connections

Derivatives of eˣ and aˣ — proofs — woh parent result jise yeh page visually derive karta hai.
Limits — first principles definition of derivative — Step 1 ka rise-over-run engine.
Chain Rule — Step 5 ka outer-slope-times-inner-slope tool.
Natural Logarithm ln x — $ln a$ matlab "woh power $e$ par jo $a$ banata hai."
The number e — definitions — same $e$ ka $(1 + \frac{1}{n})^{n}$ wala view.
Exponential Growth and Decay — Step 6 ki sign story real world mein.
Power Rule — woh rule jo yahan use NAHI karna (moving base vs moving exponent).