Foundations — Product rule — proof

4.1.14 · D1 · Maths › Calculus I — Limits & Derivatives › Product rule — proof

Yeh page assume karta hai ki aap kuch nahi jaante. Parent statement $(uv{)}^{\prime }=u^{\prime }v+u{v}^{\prime }$ ko padhne se pehle, hume us line ka har ek mark samajhna hoga. Hum ek ek symbol pe jaayenge, aur har ek ka plain meaning, ek picture, aur ek reason batayenge ki topic ko woh kyun chahiye.

1. Ek function — symbol $u(x)$

Letter $u$ bas machine ka ek naam hai (jaise kisi dog ko "Rex" bulana). $x$ brackets mein woh number hai jo aap daaalte hain. Toh $u(x)=x^2$ matlab hai "woh machine jo aapki di hui cheez ko square kar de": $3$ daalo, $9$ milega.

Topic ko yeh kyun chahiye. Parent note ek aise rectangle ki baat karta hai jiska width $x$ ke saath change hota hai. "Ek width jo $x$ pe depend kare" exactly ek function hai — hum isse $u(x)$ kehte hain. Height ek aur function hai, $v(x)$.

Picture dekho: jaise input $x$ neeche slide karta hai, machine $u$ vertical axis pe ek height produce karti hai. Woh curve hi function hai.

2. Ek tiny nudge — symbol $h$

Hum $u(x+h)$ mein interested hain: machine ka output uske baad jab hum input ko $h$ se nudge karein. Agar $h$ chota hai, toh $x+h$, $x$ ke kareeb hai, isliye $u(x+h)$, $u(x)$ ke kareeb hai — lekin exactly equal nahi, kyunki machine ne nudge pe react kiya.

Topic ko yeh kyun chahiye. Poori proof yeh poochti hai: "jab main $x$ ko ek baal $h$ se nudge karta hoon, toh area kitna change hota hai?" Aap change ke baare mein nahi pooch sakte bina ek nudge ke jisse change ho sake. Woh nudge hai $h$.

3. Output kitna hila — symbol $\Delta u$

$\Delta u$ ek number hai: machine ka answer kitna jump kiya jab hum input ko $h$ se step kiya.

Figure mein red vertical gap dekho — woh gap hi $\Delta u$ hai. Jab $h$ chota hota hai toh gap chota hota hai; $h$ ko zero ki taraf shrink karo aur gap bhi shrink hota hai.

Topic ko yeh kyun chahiye. Parent ke rectangle mein, right strip ki area $v\Delta u$ hai aur top strip ki area $u\Delta v$ hai. Poori geometric story $\Delta u$ aur $\Delta v$ mein batayi jaati hai.

4. Rate of change — derivative $u^{\prime }$

Ab poore chapter ka star.

Chota tick mark ${}^{\prime }$ poori notation hai: $u^{\prime }$ matlab "$u$ ka derivative". Aur do marks jo aap miloge:

• $\frac{\Delta u}{h}$ ko difference quotient kehte hain — ek step mein average steepness.
• Jaise $h$ shrink hoti hai, woh average instantaneous steepness ban jaata hai.

Do points ke beech blue line average slope $\Delta u/h$ hai. Jaise dusra point pehle ki taraf slide karta hai (yellow arrows), line tip ho jaati hai green tangent mein — steepness bilkul $x$ par. Woh limiting slope hai $u^{\prime }(x)$. Yeh picture parent proof ka poora engine hai; poori build ke liye Limit definition of the derivative dekho.

Topic ko yeh kyun chahiye. Product rule ek statement hai derivatives ke baare mein: yeh bataata hai $u^{\prime }$ aur $v^{\prime }$ kaise $(uv{)}^{\prime }$ mein combine hote hain. Derivative ke bina prove karne ke liye kuch nahi hai.

5. Limit — symbol $\underset{h\to 0}{\lim }$

Hum $\frac{\Delta u}{h}$ mein simply $h=0$ set nahi kar sakte, kyunki zero se divide karna forbidden hai. Limit honest tarika hai yeh poochne ka "yeh ratio kahan ja raha hai?" jabki $h$ abhi bhi zero se thoda oopar hai.

Topic ko yeh kyun chahiye. Proof mein har derivative ek limit hai. Aur proof mein ek key move, ${\lim }_{h\to 0}u(x+h)=u(x)$, ek limit statement hai jo sirf isliye kaam karta hai kyunki $u$ continuous hai — Differentiability implies continuity dekho.

6. Continuity — "koi sudden jump nahi"

Ek aisi curve ki picture socho jise aap pen uthaye bina draw kar sako. Wahi continuity hai. Ek staircase with a jump, jump pe continuous nahi hai.

Topic ko yeh kyun chahiye. Proof mein, ek factor $u(x+h)$ hai, aur hum $h\to 0$ karte hain yeh umeed mein ki woh $u(x)$ ban jaaye. Woh umeed tabhi justified hai agar $u$ continuous ho. Khushkismati se, differentiable hona continuity force karta hai, toh hume yeh free mein milti hai — lekin parent note theek karta hai ki hum isse naam dein.

7. Do functions ka product — symbol $uv$

Agar $u(x)$ ek width hai aur $v(x)$ ek height hai, toh $u(x)v(x)$ unse bane rectangle ki area hai. Woh single product woh object hai jise poora topic differentiate karta hai.

Figure dikhata hai kyun area change ke chaar pieces hain: original block, right strip $v\Delta u$, top strip $u\Delta v$, aur tiny corner $\Delta u\Delta v$. Product rule jo kuch kehta hai woh sab is picture mein rehta hai.

8. Multiply out karna — algebra jis pe hum lean karte hain

Do choti algebra habits jo proof use karti hai:

Prerequisite map

Equipment checklist

Connections

• Limit definition of the derivative — jahan se $u^{\prime }$ aur limit aate hain.
• Differentiability implies continuity — kyun ${\lim }_{h\to 0}u(x+h)=u(x)$ allowed hai.
• Power rule — pehla rule jo yeh foundations bootstrap karte hain.
• Quotient rule — division ke liye wahi machinery.
• Chain rule — composition ke liye wahi machinery.
• Leibniz rule (nth derivative of a product) — parent ka grand generalisation.