Derivative from first principles — difference quotient definition: Question bank

4.1.10 · D5 · Maths › Calculus I — Limits & Derivatives › Derivative from first principles — difference quotient defin

Yeh parent note ka conceptual companion hai. Yahan kuch bhi heavy algebra nahi chahiye; yeh sab symbols ke meaning ke baare mein hai.

Galti dhundho

Har line mein ek galat step hai. Galti ka naam batao.

f (x) = x^{2}

, toh

f (x + h) = x^{2} + h

." Kya galat hua?

Sirf pehla

x

replace kiya gaya. Har

x

block

(x + h)

ban jaata hai:

f (x + h) = (x + h)^{2} = x^{2} + 2 x h + h^{2}

\frac{2 x h + h ^{2}}{h} = 2 x h + h

." Kya galat hua?

Sirf ek term ko

h

se divide kiya gaya. Dono terms divide karo:

\frac{2 x h}{h} + \frac{h ^{2}}{h} = 2 x + h

\frac{x + h - x}{h}

mein upar koi

h

nahi hai, toh limit exist nahi karti." Kya galat hua?

h

chhupa hua hai. Conjugate se multiply karo usse surface karne ke liye:

\frac{( x + h ) - x}{h ( x + h + x )} = \frac{1}{x + h + x}

, jiska limit theek se exist karta hai.

f (x) = 1/ x

ke liye hume

\frac{- h}{x ( x + h )}

milta hai, aur

h

se divide karne par bhi

\frac{- h}{x ( x + h )}

hi rehta hai." Kya galat hua?

h

se divide karne par

h

cancel ho jaata hai:

\frac{1}{h} \cdot \frac{- h}{x ( x + h )} = \frac{- 1}{x ( x + h )}

. Numerator mein

h

ko cross out karna zaroori hai.

"Kyunki do points merge ho jaate hain, run

0

ho jaata hai, isliye hum kabhi slope define nahi kar sakte." Kya galat hua?

Points kabhi actually merge nahi hote —

h

0

ke paas jaata hai lekin nonzero rehta hai, toh run poori limit ke dauran nonzero rehta hai. Limit trend capture karti hai, forbidden endpoint ko nahi.

lim_{h \to 0} (2 x + h) = 2 x + 0 = 2 x

, lekin sirf isliye kyunki humne

h = 0

plug kiya." Wording mein kya subtly galat hai?

Substitution shortcut legal hai sirf isliye kyunki

2 x + h

ab ek continuous, well-defined expression hai (koi

h

denominator mein nahi). Yeh Step 4 ka safe re-plug hai, illegal Step-1 plug nahi.

Why questions

Boldo kyun, us sense mein jo topic demand karta hai.

Hum ek second point

x + h

par kyun introduce karte hain instead of directly ek point "at" calculus use karne ke?

Slope rise-over-run ke roop mein define hota hai, jiske liye change chahiye; ek point koi change nahi deta. Buddy point ek run manufacture karta hai jise hum baad mein shrink kar sakte hain. Dekho Average vs Instantaneous Rate of Change.

Limit lene se pehle

h

ko factor karke cancel kyun karna padta hai?

Cancel karne se pehle,

h = 0

plug karne par

\frac{0}{0}

milta hai, jisme koi information nahi hoti. Cancelling shared zero ko remove karta hai aur andar chhupe finite limit ko reveal karta hai — yahi Indeterminate forms 0 over 0 ka poora point hai.

Secant line

h \to 0

hone par tangent kyun "ban jaati" hai?

Jab buddy point

x

ki taraf slide karta hai, cutting line

(x, f (x))

ke aas-paas pivot karti hai jab tak yeh curve ko sirf touch na kare. Uska slope, difference quotient, tangent ke slope par settle ho jaata hai — derivative.

Derivative ko instantaneous rate of change kyun kaha jaata hai jabki yeh do points use karta hai?

Do points ek scaffold hain; limit unke beech ka gap mita deti hai, toh surviving number ek single instant par rate describe karta hai, kisi interval par nahi. Dekho Average vs Instantaneous Rate of Change.

Conjugate trick sirf square-root (aur similar) expressions ke liye kyun kaam karti hai?

Yeh

(a - b) (a + b) = a - b

exploit karta hai, jo roots ke ek stubborn difference ko ek saadha difference mein badal deta hai jisme woh

h

hota hai jise hume cancel karna hai. Roots ke bina conjugate ke rationalize karne ke liye kuch nahi hota.

Ek straight line ka first-principles slope

h

par kyun depend nahi karta?

Ek line ki rise exactly uske run ke proportional hoti hai, toh

\frac{rise}{h}

constant slope

m

tak simplify ho jaata hai koi bhi limit lene se pehle — har secant hi tangent hai.

Edge cases

Boundary aur degenerate inputs jinhe algorithm ko survive karna chahiye.

f (x) = 1/ x

ka difference quotient

x \to 0

hone par kya hota hai?

f^{'} (x) = - 1/ x^{2}

- \infty

tak blow up ho jaata hai; function

x = 0

par define bhi nahi hai, toh wahan koi derivative exist nahi karti. Formula silently

x = 0

ko exclude kar deta hai.

f (x) = x

ke liye

f^{'} (0)

kya hai, aur yeh special kyun hai?

f^{'} (x) = \frac{1}{2 x} \to + \infty

jab

x \to 0^{+}

; tangent vertical hai, toh

0

par koi finite slope exist nahi karta chahe

f

wahan defined ho. Dekho Continuity and Differentiability.

Kya

h

sirf ek taraf se

0

approach kar sakta hai ek one-sided derivative ke liye?

Haan — domain ke endpoint par (jaise

x

x = 0

) sirf

h \to 0^{+}

available hota hai, jo ek one-sided limit ke through ek one-sided derivative deta hai.

f (x) = ∣ x ∣

ke liye,

x = 0

par left aur right difference quotients kya dete hain?

Right side (

h > 0

)

+ 1

deta hai, left side (

h < 0

)

- 1

deta hai; kyunki yeh agree nahi karte, two-sided limit fail ho jaati hai aur

f^{'} (0)

exist nahi karta — ek corner.

f (x) = x^{2}

ka derivative exactly vertex

x = 0

par kya hai?

f^{'} (0) = 2 (0) = 0

— slope zero hai kyunki parabola ka neeche wala hissa momentarily flat hai, geometry se match karta hai.

Agar

f (x + h) - f (x)

h \to 0

hone par nahi jaata

0

ki taraf, toh yeh kya batata hai?

f

x

par discontinuous hai; phir

\frac{nonzero}{0}

blow up ho jaata hai aur koi derivative exist nahi karti — "differentiable

\Rightarrow

continuous" ka ek concrete illustration. Dekho Continuity and Differentiability.

Kya difference quotient kabhi kisi nonzero

h

ke liye exactly derivative ke barabar hota hai?

Haan linear

f

ke liye (yeh sabhi

h

ke liye slope ke barabar hota hai); curved

f

ke liye ek specific nonzero

h

Mean Value Theorem se match kar sakta hai, lekin generally quotient sirf derivative ke paas jaata hai. Dekho Average vs Instantaneous Rate of Change.

Connections

Derivative from first principles — difference quotient definition — woh parent jise yeh bank drill karta hai.
Limits — formal definition and one-sided limits — one-sided limits edge cases ko power karte hain.
Indeterminate forms 0 over 0 — wajah ki cancellation substitution se pehle honi chahiye.
Secant and Tangent lines — true/false items ke peeche ki geometry.
Continuity and Differentiability — corners aur vertical tangents.
Average vs Instantaneous Rate of Change — secant vs. tangent framing.

Question bank — Derivative from first principles — difference quotient definition

True ya false — justify karo