Foundations — Sum, difference, constant multiple rules

4.1.13 · D1 · Maths › Calculus I — Limits & Derivatives › Sum, difference, constant multiple rules

Sum, difference, aur constant-multiple rules par trust karne se pehle, aapko har ek symbol mein fluent hona chahiye jinpar ye depend karte hain. Yeh page har ek cheez ko zero se build karta hai — plain words, phir ek picture, phir kyun topic ko yeh chahiye. Upar se neeche padho: har block sirf wahi use karta hai jo pehle aa chuka hai.

1. Ek function $f(x)$ — ek machine jo numbers ko numbers mein badalta hai

Picture. Ek vending machine socho. Tum ek button dabaate ho ($x$), tumhe ek snack milta hai ($f(x)$). Same button hamesha same snack deta hai — yeh "exactly ek output" wala part matter karta hai.

Topic ko yeh kyun chahiye. Parent note baar baar $f(x)+g(x)$ ya $3x^4-5x^2+7$ jaisi cheezein likhta rehta hai. Ye bas functions hain. Aap "curve ka slope" ke baare mein tab tak baat nahi kar sakte jab tak yeh agree na ho ki curve ek function ke output ko graph ke roop mein draw kiya gaya hai.

2. Slope — ek straight line kitni steep hai

Picture. Ek line par do points chuno. Right ki taraf kuch amount walk karo (run). Count karo kitna upar (ya neeche) chade (rise). Divide karo. Ek steep hill mein chhoti run ke liye badi rise hoti hai — bada slope. Ek flat road mein zero rise hai — zero slope. Downhill ek negative slope deta hai.

Topic ko yeh kyun chahiye. "Derivative" ka matlab hoga ek curve ka slope. Lekin ek curve muda hua hota hai, isliye uski steepness point to point badlati rehti hai. Pehle hum straight lines ke liye slope pakke karte hain, phir isko extend karte hain.

3. Symbols $\Delta x$ aur "change in" — ek step measure karna

Picture. Input $x$ par khade ho. $\Delta x$ size ki ek tiny step right lo, $x+\Delta x$ par land karo. Output $f(x)$ se upar (ya neeche) $f(x+\Delta x)$ tak jump karta hai. Vertical jump rise hai; step $\Delta x$ run hai.

Topic ko yeh kyun chahiye. Yeh exactly rise-over-run hai, lekin ek curve ke liye, do nearby points ke beech: $\frac{f(x+\Delta x)-f(x)}{\Delta x}=\frac{\text{rise}}{\text{run}}$ Is fraction ko difference quotient kehte hain — iska shape yaad karo, poora derivative isi par bana hai. Dekho Definition of the derivative (limit of difference quotient).

4. Limit $\lim$ — "yeh kya approach karta hai?"

Picture. Apni sideways step $\Delta x$ ko chhota se chhota karte jao. Curve par do points ek saath slide karte hain. Rise-over-run line "do points ke beech ek chord" se aage tangent ban jaati hai — woh line jo curve ko sirf ek jagah graze karti hai. Uska slope woh number hai jis taraf difference quotient approach karta hai.

Kyun yeh tool aur sirf $0$ plug karna nahi? Kyunki difference quotient literally $\Delta x$ se divide karta hai. $\Delta x=0$ set karna zero se divide karna hai. Limit hi ekmaatra honest tarika hai yeh puchhne ka ki "jab do points merge hote hain to hum kis slope ki taraf ja rahe hain?" bina kabhi zero se divide kiye.

5. Derivative $f^{\prime }(x)$ aur $\frac{d}{dx}$ — ek point par curve ka slope

Topic ko dono kyun chahiye. Parent ke proofs $f^{\prime }(x)=\lim \dots$ (Lagrange, compact) se shuru hote hain lekin rules ko $\frac{d}{dx}[f+g]$ (Leibniz, dikhata hai kaunsa variable) ke roop mein state karte hain. Tumhe pehchaanna chahiye ki ye same object hai do costumes mein.

6. Ek constant $c$ aur uska zero slope

Picture. Ek flat line kabhi rise nahi karti. Rise $=0$ kisi bhi run ke liye, to slope $=0$. Isliye $\frac{d}{dx}(7)=0$: ek flat road mein koi steepness nahi hoti.

Topic ko yeh kyun chahiye. Constant-multiple rule $\frac{d}{dx}[cf]=c{f}^{\prime }$ ko $c$ ko ek genuine constant hona chahiye taaki woh limit se "bahar aa sake" bina kisi problem ke. Aur "ek constant ka derivative $0$ hai" wala fact exactly isliye hai ki Example 1 mein $+7$ gayab ho jaata hai.

7. Functions banana: sum, difference, scalar multiple

Picture. Kisi $x$ par $f+g$ padhne ke liye: $f$ ki height dhundho, $g$ ki height dhundho, ek ko doosre par stack karo. Naye curve ki height sum hai. $c$ se scale karna har height ko $c$ se multiply karta hai.

Topic ko yeh kyun chahiye. Ye sirf teeen hi operations hain jo linearity cover karti hai. Parent ka poora message — "adding added rehti hai, scaling scaled rehti hai" — exactly inhi teen moves ke baare mein ek statement hai aur kisi ke nahi (khaas taur par do functions ki multiplication nahi; uske liye Product rule chahiye).

8. Limit laws — kyun hum ek limit ko alag kar sakte hain

Picture. Agar do quantities dono ek value par settle down karti hain, to unka running total unhi values ke total par settle hota hai — wobbles nahi lad sakti kyunki har ek apne aap khatam ho jaati hai. Har cheez ko $c$ se scale karna bas destination ko $c$ se scale karta hai.

Topic ko yeh kyun chahiye. Ye laws parent mein har ek proof ka engine hain. Jab sum-rule proof ek bade difference quotient ko do mein split karta hai, to sum law hi us split ko valid banata hai; jab constant-multiple proof mein $c$ limit se bahar nikalti hai, yeh "constant pulls out" hai. Poori details Limit laws (sum, scalar, product of limits) mein hain.

9. Linearity — ek word jo poore topic ka naam hai

Topic ko yeh kyun chahiye. Parent ka grand conclusion hai ki $\frac{d}{dx}$ ek linear operator hai — differentiation ek linear operation hai. Sum rule, difference rule, aur constant-multiple rule is ek fact ke teen visible faces hain. Abstract view ke liye Linear operators dekho aur iske integral twin ke liye Linearity of integration dekho.

Prerequisite map

Left par sab kuch derivative ko feed karta hai; derivative plus combining moves plus limit laws linearity dete hain — parent topic.

Equipment checklist

Right side cover karo aur khud ko test karo. Agar koi bhi answer shaky hai, parent note kholne se pehle us section ko dobara padho.