Visual walkthrough — Sum, difference, constant multiple rules

4.1.13 · D2 · Maths › Calculus I — Limits & Derivatives › Sum, difference, constant multiple rules

Koi bhi formula chhune se pehle, aao ek-ek word ka matlab zero se agree kar lete hain.

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Step 0 — Pehle sirf ek word define karna zaroori hai: slope

KYA. Koi function jaise $y=f(x)$ ek rule hai: usse $x$ number do, woh $y$ number deta hai. Saare pairs $(x,y)$ plot karo toh ek curve milta hai. Kisi point par slope ek hi sawaal ka jawaab hai: agar main $x$ ko thoda sa daayein nudge karun, toh $y$ kitni tezi se chadhega? Steep uphill = bada positive slope. Downhill = negative slope. Flat = zero slope.

KYUN zaroori hai. Symbol $f^{\prime }(x)$ (padho "$f$-prime of $x$") kuch bhi nahin hai siwaaye curve $f$ ka slope point $x$ par. Agar tum slope picture kar sakte ho, tum derivative samajh chuke ho — baaki sab sirf precisely measure karna hai.

PICTURE. Red line curve ko ek point par bas chhoo rahi hai; uski steepness hi $f^{\prime }(x)$ hai.

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Step 1 — Ek sum banao, aur heights ko stack hote dekho

KYA. Do curves lo, $f$ (teal) aur $g$ (plum). Ek nayi curve banao har $x$ par unki heights add karke: $h(x)=f(x)+g(x)$

KYUN. Yahi toh interest ka object hai. Hum chahte hain $h$ ka slope — lekin $h$ banayi gayi thi do jaani-pehchaani curves ko stack karke. Sawaal yeh hai ki kya woh manufacturing process slopes par bhi carry over hoti hai.

PICTURE. Chune hue $x$ par, $h$ ki burnt-orange height literally teal segment ke upar plum segment rakh ke bani hai. Do bricks stack ki hui; total height = dono ki sum.

$h(x)=\underset{\text{teal brick}}{\underbrace{f(x)}}+\underset{\text{plum brick}}{\underbrace{g(x)}}\Rightarrow \text{orange height}=\text{teal}+\text{plum}$

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Step 2 — $x$ ko nudge karo: dono bricks badhti hain, toh sum dono ki growth se badha

KYA. $\Delta x$ daayein kadam rakho. Teal brick $\Delta f=f(x+\Delta x)-f(x)$ jitni badh jaati hai. Plum brick $\Delta g=g(x+\Delta x)-g(x)$ jitni badh jaati hai. Kyunki $h$ dono ka stack hai, iski growth dono growths ka addition hai: $\Delta h=\Delta f+\Delta g$

KYUN. Yahi sum rule ka geometric dil hai. Dono bricks ke beech koi interaction nahin — ek doosre ke upar baithti hai, toh unke increases simply pile up ho jaate hain. Koi products nahin, koi cross-terms nahin.

PICTURE. Do shaded growth-strips: $\Delta f$ height ki teal strip aur $\Delta g$ height ki plum strip. Orange total-growth bar exactly utni hi lambi hai jitni dono strips milakar.

$\underset{\Delta h,\text{orange bar}}{\underbrace{h(x+\Delta x)-h(x)}}=\underset{\Delta f,\text{teal strip}}{\underbrace{f(x+\Delta x)-f(x)}}+\underset{\Delta g,\text{plum strip}}{\underbrace{g(x+\Delta x)-g(x)}}$

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Step 3 — Step se divide karo: do slopes side by side dikhti hain

KYA. Step 2 ki har cheez ko step width $\Delta x$ se divide karo: $\frac{\Delta h}{\Delta x}=\frac{\Delta f}{\Delta x}+\frac{\Delta g}{\Delta x}$

KYUN. Rise-over-run har raw growth ko slope mein badal deta hai. Hum fraction split karte hain kyunki ek single denominator ke upar ki sum ko hamesha term by term baanta ja sakta hai — yeh toh bas fractions ka kaam karne ka tarika hai: $\frac{a+b}{c}=\frac{a}{c}+\frac{b}{c}$.

PICTURE. Teen connecting lines drawn: $h$ ke liye orange, $f$ ke liye teal, $g$ ke liye plum. Unki steepnesses satisfy karti hain: orange steepness = teal steepness + plum steepness.

$\underset{\text{slope of }h\text{’s little line}}{\underbrace{\frac{\Delta h}{\Delta x}}}=\underset{\text{slope of }f\text{’s line}}{\underbrace{\frac{\Delta f}{\Delta x}}}+\underset{\text{slope of }g\text{’s line}}{\underbrace{\frac{\Delta g}{\Delta x}}}$

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Step 4 — Step ko zero tak shrink karo: slopes derivatives ban jaate hain

KYA. $\Delta x\to 0$ hone do. Har chhoti-line slope asli grazing-line slope — derivative — par collapse ho jaati hai: ${\lim }_{\Delta x\to 0}\frac{\Delta h}{\Delta x}={\lim }_{\Delta x\to 0}\frac{\Delta f}{\Delta x}+{\lim }_{\Delta x\to 0}\frac{\Delta g}{\Delta x}$

KYUN YAHI TOOL — limit, aur yeh specific limit law. Hum limit use karte hain kyunki slope "ek single point par" ke liye dono points ko merge karna padta hai; limit hi ek aisa tool hai jo $\Delta x$ ko hard zero se divide kiye bina vanish hone deta hai. Aur hum limit ko $+$ sign ke paas split kar sakte hain sirf isliye kyunki sum law for limits hai: the limit of a sum equals the sum of the limits, provided each limit exists — dekho Limit laws (sum, scalar, product of limits). Dono pieces yahan exist karte hain precisely kyunki humne assume kiya tha ki $f$ aur $g$ differentiable hain.

PICTURE. Connecting lines ka ek fan jisme $\Delta x$ shrink ho raha hai $\to 0$; teeno fans ek saath apni-apni grazing lines par converge karte hain.

$\boxed{h^{\prime }(x)=f^{\prime }(x)+g^{\prime }(x)}■$

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Step 5 — Constant multiple rule: curve ko stretch karo toh slope bhi stretch hota hai

KYA. Ab add karne ki jagah, scale karo. $c$ ek fixed number ho aur $h(x)=cf(x)$: $f$ ki har height ko $c$ se multiply karo. Claim: slope bhi $c$ se multiply ho jaata hai.

KYUN. Curve ko vertically $c$ se stretch karne par har rise $c$ times bada ho jaata hai, jabki horizontal run $\Delta x$ unchanged rehta hai. Rise-over-run isliye $c$ se scale ho jaata hai. Algebraically, $c$ $\Delta x$ par depend nahin karta, toh woh limit se seedha bahar aa jaata hai — Limit laws (sum, scalar, product of limits) wali scalar law for limits.

PICTURE. $c=2$ ke liye, plum stretched curve har jagah twice as tall hai; uski grazing line twice as steep hai. $c=\frac{1}{2}$ ke liye, half as tall, half as steep.

$h^{\prime }(x)={\lim }_{\Delta x\to 0}\frac{cf(x+\Delta x)-cf(x)}{\Delta x}=c\underset{=f^{\prime }(x)}{\underbrace{\underset{\Delta x\to 0}{\lim }\frac{f(x+\Delta x)-f(x)}{\Delta x}}}=c{f}^{\prime }(x)$

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Step 6 — Difference koi naya rule nahin hai (aur jo sign trap yeh chhupata hai)

KYA. Subtraction ek negative ka addition hai: $f-g=f+(-1)g$. Step 5 apply karo $c=-1$ ke saath $g$ par (use ulta palat do), phir Step 5 ka sum rule: $\frac{d}{dx}[f-g]=f^{\prime }+(-1)g^{\prime }=f^{\prime }-g^{\prime }$

KYUN. Koi nayi geometry nahin chahiye — $g$ ko ulta palatne se uske slope ka sign flip ho jaata hai, phir heights (ab ek negative hai) phir bhi stack ho jaati hain. Yahi "difference is free" idea hai, visible banaya gaya.

PICTURE. Plum $g$ horizontal axis ke through reflect ho kar $-g$ bana; iski downhill slope uphill ho jaati hai. Teal $+$ flipped-plum stack karne par $f-g$ milta hai.

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Step 7 — Edge & degenerate cases (koi bhi scenario kabhi undiscussed mat chhodna)

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Ek-picture summary

Ek frame: teal $f$ aur plum $g$ orange $h=f+g$ mein stack kiye, teeno grazing lines drawn, aur chhota slope-triangle dikhata hai orange-tilt = teal-tilt + plum-tilt. Woh single visual hi linearity of the derivative hai — aur iska abstract naam Linear operators par milta hai, integral twin ke saath Linearity of integration par.

Recall Feynman retelling — poora walkthrough seedhe shabdon mein

Socho do pahaad hain, ek teal aur ek plum, aur tum ek naya pahaad banate ho har point par plum pahaad ki height ko teal pahaad ke upar rakh ke. Ab ek chhota sa kadam aage lo. Teal zameen thodi si oopar aayi; upar rakhi plum pile thodi aur aayi. Tumhari total chadhaan bas woh do chhoti chadhaan add hui — piles kabhi ek doosre se nahi ladte. Har chadhaan ko apne kadam ki lambai se divide karo aur tumhare paas slopes hain; kadam ko zero tak shrink karo aur slopes derivatives ban jaate hain. Toh combined pahaad ki steepness exactly dono steepnesses add hui hai: yahi sum rule hai. Ab agar tumne ek pahaad ko twice as tall stretch kiya hota, toh har rise double ho jaata jabki tumhara stride same rehta, toh uski steepness bhi double ho jaati — yahi constant multiple rule hai. Aur "$g$ par neeche jaana" $g$ ko ulta palat ke add karne jaisa hai, jo subtraction hai — koi naya rule nahin, bas ek minus sign jo tum poore doosre pahaad par apply karte ho. Addition added rehta, scaling scaled rehta. Bas itna hi.

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Connections

• Sum, difference, constant multiple rules — woh parent jise yeh page draw karti hai.
• Definition of the derivative (limit of difference quotient) — Step 0 ki slope machine.
• Limit laws (sum, scalar, product of limits) — sum law (Step 4) aur scalar law (Step 5).
• Power rule — linearity ke saath pair karo aur kisi bhi polynomial ko term by term differentiate karo.
• Product rule — non-linear sibling: heights multiply hoti hain, aur stacking logic toot jaata hai.
• Linearity of integration — wahi "stacks stay stacked" picture, ulti direction mein chali.
• Linear operators — "add stays added, scale stays scaled" ka abstract ghar.