Sum, difference, constant multiple rules: Question bank

4.1.13 · D5 · Maths › Calculus I — Limits & Derivatives › Sum, difference, constant multiple rules

Definition Woh vocabulary jis par yeh page chalta hai

Pehle kuch words, taaki koi cheez use hone se pehle bani na ho.

Derivative $f^{'} (x)$ — curve $y = f (x)$ ka $x$ point par slope: graph wahan kitna steep hai.
Difference quotient — curve ke do kareeb points ko jodne wali chhoti line ka slope, ek $x$ par aur ek thoda sa aage. Us chhote horizontal step ko hum == $Δ x$ == kehte hain (padho "delta-x"), bas $x$ mein ek chhota sa change. Difference quotient hai $\frac{f ( x + Δ x ) - f ( x )}{Δ x}$ (rise over run). $Δ x$ ko $0$ ki taraf shrink karna is line ko sachchi slope mein badal deta hai — wahi hai derivative.
Linear — ek rule linear hai agar woh yeh teeno kaam kare:
1. sum ke paas split ho jaye: $[f + g]^{'} = f^{'} + g^{'}$
2. difference ke paas split ho jaye: $[f - g]^{'} = f^{'} - g^{'}$
3. constant factor ko bahar nikalne de: $[c f]^{'} = c f^{'}$

(==...== marks sirf is vault ka tarika hai key term ko flashcards ke liye flag karne ka — inhe highlighter samjho, maths nahi.)

Neeche ki do pictures poora engine hain in rules ke peeche. Page par har trap actually inhi pictures mein se kisi ek ki galat reading hai — jab bhi koi answer kahe "slopes add hote hain" ya "constant saath chala jaata hai" toh inhe dobara dekh lo.

Figure — Sum, difference, constant multiple rules

True ya false — justify karo

Kisi bhi do differentiable functions ke liye, ek sum ki derivative, derivatives ka sum hoti hai.

True — yeh exactly sum rule hai

\frac{d}{d x} [f + g] = f^{'} + g^{'}

. Picture s01: combined curve ka kisi bhi step par rise over run sirf

f

ka rise plus

g

ka rise hai, toh slopes stack hote hain; formally limits ka sum law difference quotient ko do mein split karne deta hai.

Product ki derivative, derivatives ka product hoti hai.

False — yeh classic over-generalisation hai. Linearity sirf

+, -, \times constant

cover karti hai; products ko Product rule chahiye

(f g)^{'} = f^{'} g + f g^{'}

. Test:

\frac{d}{d x} (x \cdot x) = 2 x

lekin

x^{'} \cdot x^{'} = 1

\frac{d}{d x} [5 f (x)] = 5 f^{'} (x)

har constant aur har differentiable

f

ke liye.

True — constant multiple rule; woh

5

kabhi bhi

Δ x

(chhota step) par depend nahi karta, toh difference quotient mein woh factor out ho jaata hai aur seedha limit se baar nikal jaata hai bina kisi change ke.

\frac{d}{d x} [f (x) \cdot 5] = f^{'} (x) \cdot 5

constant multiple rule se alag rule hai.

False — yeh wahi rule hai. Constant se multiply karna commutative hai, toh

5 f

aur

f \cdot 5

identical hain, aur dono

5 f^{'}

dete hain.

Agar

c

x

par depend karta ho — maano

c = x

— toh

\frac{d}{d x} [c f] = c f^{'}

phir bhi hold karta hai.

False — "constant" ka matlab literally constant hona chahiye. Agar

c = x

hai toh

\frac{d}{d x} [x f]

ko product rule chahiye, jo

f + x f^{'}

deta hai, na ki

x f^{'}

f (x) - g (x)

ki derivative ke liye limit definition se alag proof chahiye.

False — kyunki

f - g = f + (- 1) g

hai, sum rule plus constant multiple rule (

c = - 1

ke saath) ise free mein de deta hai.

Linearity teen ya zyada terms ke liye ek saath kaam karti hai, jaise

\frac{d}{d x} [f + g + h]

True — sum rule ko baar baar apply karo (

(f + g) + h

ki tarah group karo). Linearity kisi bhi finite number of terms mein kisi bhi scalar coefficients ke saath extend hoti hai.

\frac{d}{d x} [2 sin x + 3 e^{x}]

ko difference quotient par dobara gaye bina compute kiya ja sakta hai.

True — yahi toh linearity ka poora faida hai: har known piece ko differentiate karo aur recombine karo, use ke point par koi limits ki zaroorat nahi.

Galti dhundho

\frac{d}{d x} (x^{2} - 3 x + 4) = 2 x - 3 + 4

." — galti dhundho.

Constant

4

ki derivative

0

hoti hai,

4

nahi. Sahi:

2 x - 3

. Ek constant flat line hai, toh uska slope zero hota hai.

\frac{d}{d x} [f - (x^{2} - 3 x)] = f^{'} - 2 x - 3

." — galti dhundho.

Minus sign poore bracket par distribute hona chahiye. Ise

(- 1) (x^{2} - 3 x)

samjho: derivative hai

- (2 x - 3) = - 2 x + 3

, toh jawab hai

f^{'} - 2 x + 3

\frac{d}{d x} [x^{2} \cdot sin x] = 2 x \cdot cos x

." — galti dhundho.

Products split nahi hote. Yahan galti se sum-style rule ko product par apply kiya gaya. Product rule use karo:

2 x sin x + x^{2} cos x

\frac{d}{d x} [\frac{x}{5}] = \frac{1}{5 x}

." — galti dhundho.

\frac{x}{5}

asal mein

\frac{1}{5} x

hai, ek constant

\frac{1}{5}

times

x

, toh iski derivative hai

\frac{1}{5} \cdot 1 = \frac{1}{5}

. Student ne ise

\frac{1}{5 x} = \frac{1}{5} x^{- 1}

samajh liya.

\frac{d}{d x} [7 f (x)] = 7 + f^{'} (x)

." — galti dhundho.

Ek constant

f

ko multiply karta hai, add nahi karta. Constant multiple rule

7 f^{'} (x)

deta hai, na ki

7 + f^{'} (x)

"Kyunki

\frac{d}{d x} [c f] = c f^{'}

hai, toh

\frac{d}{d x} [c + f] = c + f^{'}

." — galti dhundho.

Joda gaya constant

c

ki derivative

0

hoti hai, toh

\frac{d}{d x} [c + f] = f^{'}

hai, na ki

c + f^{'}

. Constant add karna graph ko upar shift karta hai lekin kabhi tilt nahi karta, toh slope unchanged rehti hai.

Why questions

Sum rule limit definition se kyun nikalti hai lekin product rule nahi?

s01 dekho: numerator

[f (x + Δ x) + g (x + Δ x)] - [f (x) + g (x)]

regroup hokar ek

f

-difference plus ek

g

-difference banta hai — do saaf difference quotients. Ek product

f (x + Δ x) g (x + Δ x)

ki jagah ek cross-term rehti hai jo separate hone se mana karti hai, jiski wajah se extra

f^{'} g + f g^{'}

structure force hota hai.

Constant multiple proof mein constant

c

ko limit ke bahar kheenchna legitimate kyun hai?

Kyunki

c

Δ x

(shrinking step) par depend nahi karta, toh jab

Δ x \to 0

hota hai woh fixed rehta hai; limit law "constant factors limits se bahar aa jaate hain" exactly apply hota hai. s02 mein poori curve vertically

c

se stretch hoti hai, toh har slope bhi

c

se scale hoti hai.

Sum rule hold karne ke liye

f

aur

g

dono differentiable kyun hone chahiye?

Limits ke sum law ko dono limits (dono slopes) exist karne chahiye pehle tum unhe split kar sako. Agar

f^{'}

g^{'}

mein se koi exist nahi karta, toh split justify nahi hoti — bhale hi special cases mein sum phir bhi differentiable ho sakta hai (neeche wala last edge case dekho).

\frac{d}{d x}

ko "linear operator" kehna sirf ek slogan se zyada kyun hai?

Kyunki yeh linearity ki do defining properties satisfy karta hai — additivity aur scalar homogeneity — jo

\frac{d}{d x} [a f + b g] = a f^{'} + b g^{'}

mein captured hain. Yeh ise doosre Linear operators ke saath rakhta hai aur Linearity of integration ko mirror karta hai.

Difference rule ko apne limit-law ki zaroorat kyun nahi?

Difference asal mein negative ka addition hai:

f - g = f + (- 1) g

. Toh wahi do ingredients (sum law + constant bahar aata hai, dono s01 aur s02 mein pictured) ise already cover karte hain — subtraction kuch naya introduce nahi karti.

Linearity ek derivative mein sine, exponential, aur logarithm ko bina kisi interaction ke kyun mix kar sakti hai?

Kyunki har term ek scalar times ek function hai jo doosron mein add hoti hai; linearity har summand ko independently treat karti hai (har ek s01 stack mein apni layer hai). Interaction tabhi aati hai jab functions multiply ya compose hote hain, jo linearity govern nahi karti.

Edge cases

\frac{d}{d x} [0 \cdot f (x)]

kya hai?

Yeh

0 \cdot f^{'} (x) = 0

hai — constant multiple rule ka ek valid instance jahan

c = 0

hai. Function

x

-axis par flatten ho jaata hai, toh iski slope har jagah zero hai.

\frac{d}{d x} [f (x) - f (x)]

kya hai?

Difference rule

f^{'} (x) - f^{'} (x) = 0

deta hai, jo is fact ke saath consistent hai ki

f - f

constant function

0

hai. Dono raaste agree karte hain — ek achha sanity check.

Kya linearity

\frac{d}{d x} [f (x) /2]

aur

\frac{d}{d x} [2/ f (x)]

ke baare mein kuch kehti hai?

Pehli

\frac{1}{2} f^{'} (x)

hai (constant multiple, kyunki

/2 = \times \frac{1}{2}

). Doosri mein denominator mein variable

f (x)

hai, toh yeh constant multiple nahi hai aur quotient/chain rules chahiye.

Agar

g (x)

differentiable hai lekin

f (x)

nahi, toh kya

\frac{d}{d x} [f + g]

phir bhi exist kar sakta hai?

Haan, principle mein sum differentiable ho sakta hai bhale ek piece nahi ho — lekin sum rule apply nahi hoga, kyunki iske liye limit split karne ke liye dono slopes ka exist karna zaroori hai. Tumhe sum ko directly analyse karna hoga.

Do functions jo dono non-differentiable hain, ek smooth function mein sum ho sakti hain — kaise? (Figure dekho.)

f (x) = ∣ x ∣

aur

g (x) = - ∣ x ∣

lo: dono ka

x = 0

par ek sharp corner hai jahan koi single slope exist nahi karti. Lekin

f (x) + g (x) = 0

ek flat line hai, bilkul differentiable with derivative

0

. Picture s03: dono corners mirror images hain aur cancel ho jaate hain. Sabak — tum yahan sum rule invoke nahi kar sakte (na

f^{'} (0)

exist karta hai na

g^{'} (0)

), bhale hi sum differentiable ho. Pehle dono pieces check karo.

Negative constant ke liye constant multiple rule kya deta hai, jaise

\frac{d}{d x} [- 4 f (x)]

Yeh

- 4 f^{'} (x)

deta hai; sign bhi kisi bhi doosre constant ki tarah saath chali jaati hai. Yahi mechanism (

c = - 1

) hai jo difference rule generate karta hai.

Kya

\frac{d}{d x} [c] = 0

constant multiple rule ka ek special case hai?

Haan — constant

c

c \cdot 1

likho jahan function constant

1

hai; phir

c \cdot (1)^{'} = c \cdot 0 = 0

. Har case mein flat line ki slope zero hoti hai.

Connections

Sum, difference, constant multiple rules — parent note with full proofs.
Limit laws (sum, scalar, product of limits) — upar har split ko justify karta hai.
Definition of the derivative (limit of difference quotient) — ground floor.
Power rule — linearity ke saath pair karke kisi bhi polynomial ko differentiate karta hai.
Product rule — woh non-linear sibling jis par yeh traps baar baar point karte hain.
Linearity of integration · Linear operators — wahi structure, kahin aur.