Derivatives of ln x and logₐ(x): Foundations — Hinglish deep dive

4.1.20 · D1 · Maths › Calculus I — Limits & Derivatives › Derivatives of ln x and logₐ(x)

Yeh page maanta hai tumne kuch nahi dekha. Isse pehle ki hum $ln x$ ke slope ke baare mein baat karein, humein page ke har ek mark ko samajhna hoga: power kya hoti hai, log kya undo karta hai, limit kya hai, $e$ kya hai, slope kya hai, aur yeh ajeeb sa squiggle $\frac{d}{d x}$ kya matlab rakhta hai. Hum inhe order mein build karenge — har block sirf upar waale blocks pe lean karta hai.

1. Powers — shuru ka brick

Isko picture karo. $2^{3}$ teen copies of $2$ hain jo product mein stack hain: $2 \times 2 \times 2 = 8$ . Jaise exponent badhta hai, value tezi se upar jaati hai.

Figure — Derivatives of ln x and logₐ(x)

Figure padhein. Har bar $n = 0, 1, 2, 3, 4, 5$ ke liye $2^{n}$ hai. Notice karo ki sabse left wala bar height $1$ hai (yeh $2^{0}$ hai, neeche explain hoga) aur right mein har bar pehle wale se double hai — yeh doubling hi hai "ek aur copy of $2$ se multiply karo" kaisa dikhta hai, aur magenta arrow dikhata hai ki yeh kitni tezi se bhaagta hai.

Definition Teen "extended" cases — taaki koi exponent tumhe surprise na kare

Upar ki picture sirf $n = 1, 2, 3, \dots$ cover karti hai. Lekin hum $a^{0}$ , negative aur fractional exponents se milenge. Poore topic mein base $== a > 0 ==$ (positive) lo taaki neeche ke har case mein real number mile. Hum new cases ko define karte hain taaki neeche wala adding-rule kabhi na toote:

$a^{0} = 1$ . Kyun forced? Hum chahte hain $a^{1} \cdot a^{0} = a^{1 + 0} = a^{1}$ . $a^{1}$ se divide karne par $a^{0} = 1$ milta hai (jabtak $a \neq = 0$ ).
$a^{- n} = \frac{1}{a ^{n}}$ . Kyun forced? Hum chahte hain $a^{n} \cdot a^{- n} = a^{0} = 1$ , isliye $a^{- n}$ reciprocal hona chahiye.
$a^{1/2} = a$ , aur iske liye $a \geq 0$ chahiye taaki $a$ real number rahe. Kyun forced? Hum chahte hain $a^{1/2} \cdot a^{1/2} = a^{1} = a$ , isliye $a^{1/2}$ ko square karne par $a$ milna chahiye. Zyada generally $a^{p / q} = (q a)^{p}$ whole numbers $p, q$ ke liye — yeh bhi sirf tabhi safe hai jab $a > 0$ .

Topic ko yeh kyun chahiye. Logs ke baare mein sab kuch powers ke upar likha hai. Tum "kaun si power mujhe yeh deti hai?" tab tak nahi pooch sakte jab tak tum nahi jaante ki power kya hai — har tarah ki power ke liye, whole, zero, negative ya fractional. Dekho Derivative of e^x and a^x ki yeh kaise functions mein grow karte hain.

2. The logarithm — exponent ko ulta poochna

Isliye log koi naya machine nahi — yeh power ka undo button hai. Neeche ki do lines ek hi baat kahti hain left-to-right aur right-to-left padhne par:

$a^{y} = x ⟺ y = lo g_{a} x .$

Isko picture karo. Ek power exponent $y$ leta hai aur ek value $x$ produce karta hai jo explosively badhti hai. Log uska mirror image hai: yeh badi value $x$ leta hai aur chota exponent $y$ padhta hai.

Figure padhein. Orange curve $y = 2^{x}$ hai; ise exponent $3$ do (bottom axis) aur yeh $8$ output karta hai (marked orange dot). Magenta curve iska undo hai, $y = lo g_{2} x$ ; ise $8$ do aur yeh $3$ return karta hai (magenta dot). Dono curves dashed navy line $y = x$ ke across exact reflections hain — woh reflection hi "inverse / undo" geometrically kaisa dikhta hai.

Hum yeh bhi insist karte hain ki $a > 0$ aur $a \neq = 1$ : agar $a = 1$ to $1^{y} = 1$ hamesha, isliye yeh kabhi kisi aur $x$ ke barabar nahi ho sakta — sawaal ka koi jawab nahi hai.

3. Limits — "woh value jis ki taraf hum ja rahe hain"

Isse pehle ki hum $e$ ya slope build karein, humein ek honest tool chahiye "zyada se zyada close aane ke liye bina zaruri pahunche." (Humne ise ek baar already use kar liya, $a^{2}$ ko define karne ke liye.)

Humein yeh kyun chahiye. Teen ideas — irrational $x$ ke liye $a^{x}$ ko meaning dena, number $e$ , aur ek curve ki slope — sab poochte hain "kya hota hai jab koi cheez shrink/grow karti hai, agar main endpoint plug in nahi kar sakta?" Limit iska safe answer hai. Yeh Standard Limit (1+t)^{1/t} → e ko underpin karta hai.

4. Number $e$ — nature ka favourite base

Yeh aata kahan se hai? Ek bank dekho jo interest pay karta hai aur use zyada se zyada baar reinvest karta hai. $\$ 1 $ses h u r u k a r o . I n t er es t e k baa r e n d m e ina ppl y k a r o :$ (1+1)^1 = 2 $. D o ha l f - p a y m e n t s m e in s pl i t k a r k eco m p o u n d k a r o :$ (1+\tfrac12)^2 = 2.25 $.$ n $p i eces m e in :$ \left(1+\tfrac1n\right)^n $.$ n $k o a u r ba d ak a r t e j a o —§3 k i l imi t u se k a r k e — a u r n u mb er c ha d haiban d k a r d e t ahai; w o h$ e$ par settle ho jaata hai:

$e = lim_{n \to \infty} (1 + \frac{1}{n})^{n}, equivalently lim_{t \to 0} (1 + t)^{1/ t} = e .$

Doosra form kyun? $t = 1/ n$ set karo. Tab " $n$ bada" ban jaata hai " $t$ chota," yani $n \to \infty$ $t \to 0$ mein badal jaata hai — bilkul wahi $lim$ ka matlab jo humne abhi define kiya. Dekho Standard Limit (1+t)^{1/t} → e.

Figure padhein. Har violet dot $(1 + 1/ n)^{n}$ hai ek badhte $n$ ke liye (horizontal axis, log-spaced). Early dots tezi se chadh'te hain, baad ke dots muskil se hilte hain, aur sab dashed magenta line ke paas press hote hain height $e \approx 2.718$ par. Woh line par flatten hona exactly " $lim = e$ " kaisa dikhta hai.

5. Slope — pahari kitni steep hai?

Isko picture karo. Ek line par do points ek right triangle banate hain: horizontal side run hai, vertical side rise hai. Slope uss triangle ki height hai per unit width.

Figure padhein. Orange curve $y = ln x$ hai. Dashed magenta line $x = 1$ par isse just kiss karti hai; chota navy triangle ek run bottom ke saath aur matching rise side pe dikhata hai — unka ratio slope hai. Notice karo ki right mein violet note: wahan curve almost flat hai, isliye uska rise-per-run (slope) chota hai.

Lekin $ln x$ ek curved pahari hai — uski steepness har point par alag hai. Isliye humein ek aur idea chahiye: ek single point par slope.

Hum ise kaise compute karte hain. $h$ ko $x$ mein ek small change maano — $x$ se $x + h$ tak sideways step kitna hai. Tab rise $f (x + h) - f (x)$ hai aur run $h$ hai, isliye uss chote step ka rise-over-run $\frac{f ( x + h ) - f ( x )}{h}$ hai. Ab §3 ki limit use karke run ko kuch nahi tak squeeze karo:

$f^{'} (x) = \frac{d y}{d x} = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h} .$

Limit kyun? Agar hum exactly zero ka run use karein to hum zero se divide karenge — forbidden. Isliye hum dekhte hain ki rise-over-run kahan jaata hai jab step $h$ shrink karta hai. Woh "jaana" exactly $lim_{h \to 0}$ capture karta hai.

6. Do facts jo humein earn karne chahiye: $\frac{d}{d x} e^{x} = e^{x}$ , phir $\frac{d}{d x} ln x$

Parent topic $ln x$ ke slope par hinge karta hai. Ise honestly paane ke liye pehle hum dikhate hain kyun exponential $e^{x}$ apni khud ki slope hai, sirf limit tool aur $e$ ki definition use karke.

6a. Kyun $e^{x}$ khud mein differentiate hota hai

Derivative-as-limit ko $f (x) = e^{x}$ par apply karo:

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

Humne abhi kya kiya: $e^{x}$ ke liye rise-over-run likha. Kyun: yeh slope ki single definition hai jo hamare paas hai.

Power-law $e^{x + h} = e^{x} \cdot e^{h}$ use karo (§1 se) aur common $e^{x}$ bahar nikalo — yeh $h$ par depend nahi karta:

$= lim_{h \to 0} \frac{e ^{x} e ^{h} - e ^{x}}{h} = e^{x} \cdot lim_{h \to 0} \frac{e ^{h} - 1}{h} .$

Yeh step kyun: $e^{x}$ factor karna ek constant (limit ke nazariye se) ko us ek piece se alag karta hai jo abhi bhi move kar rahi hai.

Ab poora sawaal yeh hai: $h \to 0 lim \frac{e ^{h} - 1}{h}$ kya hai? Dekho ki tiny $h$ ke liye $e^{h} - 1$ kaisa dikhta hai. $e = lim_{t \to 0} (1 + t)^{1/ t}$ (§4) se, tiny $t$ ke liye $e^{t} \approx 1 + t$ , isliye $e^{t} - 1 \approx t$ , hence $\frac{e ^{t} - 1}{t} \to 1$ :

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

Yeh kaisa dikhta hai: curve $y = e^{x}$ $y$ -axis ko height $1$ par exactly slope $1$ ke saath cross karta hai — woh "slope $= 1$ shuru mein" hi $1$ kahan se aaya. Back substitute karo:

$\frac{d}{d x} e^{x} = e^{x} \cdot 1 = e^{x} .$

Isliye $e^{x}$ woh ek function hai jiska slope har point par apni khud ki height ke barabar hai. Yeh exactly woh special property hai jo humne §4 mein base $e$ ke liye announce ki thi — ab prove ho gayi. (Yeh Derivative of e^x and a^x mein aur develop kiya gaya hai.)

6b. Uss se, $ln x$ ki slope — har algebra step dikhaya gaya

Hum $\frac{d}{d x} ln x$ chahte hain. Defining relation se shuru karo, kisi formula se nahi. Maano $y = ln x$ , jo §2 ke hisaab se matlab hai

$e^{y} = x .$

Humne abhi kya kiya: $ln$ ko us exponent ke roop mein rewrite kiya jo $x$ par land karta hai. Kyun: hum $e^{something}$ differentiate karna jaante hain, lekin $ln$ directly nahi.

Ab dono sides ko $x$ ke respect mein differentiate karo. Left side $e^{y}$ hai jahan $y$ khud $x$ ka function hai; right side sirf $x$ hai:

$\frac{d}{d x} (e^{y}) = \frac{d}{d x} (x) .$

Right side $1$ hai (slope- $1$ line). Left side ko chain rule (§7) chahiye: outside $e^{(\cdot)}$ hai, jiska derivative §6a ke hisaab se wahi hai, times inside $y$ ka derivative:

$e^{y} \cdot \frac{d y}{d x} = 1.$

Humne abhi kya kiya: "outside derivative times inside derivative." Kyun: $y$ $x$ par depend karta hai, isliye bare $e^{y}$ mein ek inner change chupi hai jise humein account karna hai. Yeh kaisa dikhta hai: height $e^{y}$ dono isliye move karta hai kyunki $e$ steep hai aur kyunki $y$ khud shift ho raha hai.

$\frac{d y}{d x}$ ke liye solve karo dono sides ko $e^{y}$ se divide karke, phir $e^{y}$ ko $x$ se replace karo (woh equal hain, top line se):

$\frac{d y}{d x} = \frac{1}{e ^{y}} = \frac{1}{x} .$

$\frac{d}{d x} ln x = \frac{1}{x}, x > 0.$

7. Do tools jo upar ride karte hain

Topic ko yeh kyun chahiye. Real problems mein $ln (3 x^{2} + 1)$ hota hai, bare $ln x$ nahi. Inside $u = 3 x^{2} + 1$ ko account karna padta hai, jo $\frac{u ^{'}}{u}$ deta hai. Humne exactly yeh rule §6b mein $e^{y}$ differentiate karte waqt use kiya.

Topic ko yeh kyun chahiye. Change-of-base mein, $lo g_{a} x = \frac{ln x}{ln a}$ , denominator $ln a$ ek fixed number hai — yeh factor out ho jaata hai, baaki rehta hai $\frac{1}{ln a} \cdot \frac{1}{x} = \frac{1}{x ln a}$ . Yahi hai $\frac{d}{d x} lo g_{a} x$ ki poori derivation (deeper uses ke liye Logarithm Laws aur Logarithmic Differentiation dekho).

8. Yeh sab topic ko kaise feed karta hai

Equipment checklist

Right side cover karo aur khud se test karo. Agar koi bhi answer surprise kare, parent note se pehle woh section dobara padho.

a^{n}

ka kya matlab hai jab

n

ek positive whole number hai?

Base

a

ko khud se

n

baar multiply karo.

a^{0} = 1

kyun hai?

Kyunki

a^{1} \cdot a^{0} = a^{1}

a^{0} = 1

forced hota hai (adding-rule hold karni chahiye).

a^{- 2}

kya hai?

1/ a^{2}

— reciprocal, taaki

a^{2} \cdot a^{- 2} = a^{0} = 1

ho sake.

a^{1/2}

(aur general real powers) ke liye base

a

positive kyun hona chahiye?

Taaki

a

aur sabhi root/limit values real numbers rahein.

a^{x}

ko meaning kaise dein jab

x

irrational ho (jaise

2

x

ko fractions se squeeze karo; values

a^{fraction}

ek limit ki taraf jaati hain, jise hum

a^{x}

define karte hain.

a^{m} \cdot a^{n} = a^{m + n}

kyun?

a

m

copies ke saath

n

copies milane par product mein

m + n

copies milti hain.

Symbol

lim_{p \to q}

ka kya matlab hai?

Woh value jis ki taraf expression jaata hai jab

p

q

ki taraf push kiya jaata hai (bina

p = q

hone ki zaroorat ke).

lo g_{a} x

kaun sa sawaal answer karta hai?

a

ko kis power tak uthana padega taaki

x

mile?"

y = lo g_{a} x

ko bina log ke rewrite karo.

a^{y} = x

lo g_{a} x

ke liye

x > 0

kyun hona chahiye?

Positive base ko kisi bhi power tak uthane par hamesha positive milta hai, isliye woh kabhi

0

ya negative number ke barabar nahi ho sakta.

e

roughly kya hai aur ise define karne ka ek tarika?

e \approx 2.718

;

e = lim_{t \to 0} (1 + t)^{1/ t}

ln x

kiska short form hai, aur ise "natural" kya banata hai?

lo g_{e} x

; natural isliye kyunki

ln e = 1

y = f (x)

ka kya matlab hai?

y

woh height hai jo rule

f

input

x

se produce karta hai.

f^{'} (x)

mein prime ka kya matlab hai?

f

ka derivative (slope);

f^{'} (x) = \frac{d}{d x} f (x)

Derivative limit mein

h

kya hai?

x

mein ek small change —

x

x + h

tak sideways step.

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

kya hai, aur kyun?

1

, kyunki tiny

h

ke liye

e^{h} \approx 1 + h

, isliye

(e^{h} - 1) / h \to 1

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

kyun hai?

Factoring se milta hai

e^{x} lim_{h \to 0} (e^{h} - 1) / h = e^{x} \cdot 1

ln x

ka derivative aur uska domain batao.

\frac{d}{d x} ln x = \frac{1}{x}

for

x > 0

0 < a < 1

ke liye,

\frac{d}{d x} lo g_{a} x

positive hai ya negative?

Negative, kyunki

ln a < 0

f (u)

ke liye chain rule batao.

\frac{d}{d x} f (u) = f^{'} (u) \cdot u^{'}

ln a

jaisa constant factor derivative se kyun sirf factor out ho jaata hai?

Kyunki

\frac{d}{d x} (c f) = c f^{'}

— fixed numbers saath chale jaate hain bina change ke.