4.1.14 · D2 · HinglishCalculus I — Limits & Derivatives
Visual walkthrough — Product rule — proof
4.1.14 · D2· Maths › Calculus I — Limits & Derivatives › Product rule — proof
Shuru karne se pehle, teen simple words jo poore waqt kaam aayenge:
Neeche sab kuch ek story hai ki jab hum ko se push karte hain toh rectangle ke area ka kya hota hai.
Step 1 — Area ko rectangle ki tarah draw karo
KYA. Do changing quantities ko perpendicular sides pe rakho. Width neeche ki taraf jaati hai, height side se upar jaati hai. Unka grey box jo banta hai uska area hai — aur woh product hi function hai jise hum differentiate karna chahte hain.
KYUN. Do positive numbers ka multiplication hi rectangle ka area hota hai. Yeh koi metaphor nahi — yeh area ki definition hai. Toh agar hum samajhna chahte hain ki ek product kaise change hota hai, toh honest baat yeh hai ki dekhein ek rectangle ka area kaise change hota hai.
PICTURE.
Bottom edge ko label kiya gaya hai, left edge ko , aur shaded region pe likha hai. Abhi kuch move nahi hua — yeh "before" frame hai.
Step 2 — ko nudge karo; dono sides stretch hoti hain
KYA. ko tak advance karo. Width apni jagah nahi rehti: woh ban jaati hai. Height ban jaati hai. Naaye, bade rectangle ko purane ke upar draw karo taaki compare kar sakein.
KYUN. Derivative poochta hai "ek tiny push ko dene se kya hota hai?" ko tiny push dene se dono aur move karte hain, kyunki dono pe depend karte hain. Yahi wajah hai ki answer sirf nahi hai — do cheezein ek saath move karti hain, aur hume dono ka hisaab rakhna hoga.
PICTURE.
Original rectangle inner solid box hai; naaya outline se daayein extend hota hai (blue arrow) aur se upar (pink arrow). Naaya area hai.
Step 3 — Growth ko chaar honest pieces mein kaato
KYA. Naaya area minus purana area woh L-shaped rim hai jo humne abhi add kiya. Us rim ko teen tiles mein kaato:
\;=\; \underbrace{v\,\Delta u}_{\text{right strip}} \;+\; \underbrace{u\,\Delta v}_{\text{top strip}} \;+\; \underbrace{\Delta u\,\Delta v}_{\text{corner}} $$ Tiles ko wahaan padhte hain jahan woh hain: - $v\,\Delta u$ — **daayein taraf ki tall thin strip**: yeh $\Delta u$ wide aur $v$ tall hai. - $u\,\Delta v$ — **upar ki flat thin strip**: yeh $u$ wide aur $\Delta v$ tall hai. - $\Delta u\,\Delta v$ — **corner ka chota sa square** jahan dono strips overlap karte hain: $\Delta u$ by $\Delta v$. **KYUN.** Hume area ka *change*, $\Delta(uv)$, chahiye. $(u+\Delta u)(v+\Delta v)$ expand karke $uv$ cancel karne par exactly yahi teen tiles milte hain. Geometry aur algebra agree karte hain — multiplied-out formula aur sliced picture ek hi statement hain. **PICTURE.** ![[deepdives/dd-maths-4.1.14-d2-s03.png]] Har tile coloured hai aur uske area ke saath labelled hai. Notice karo ki corner genuinely tiny hai — ek chota number times ek chota number. > [!formula] Area mein change, exactly > $$ \Delta(uv) = v\,\Delta u + u\,\Delta v + \Delta u\,\Delta v $$ > Har term picture mein ek real area hai — abhi tak kuch approximate nahi kiya gaya. --- ## Step 4 — Corner kyun negligible hai (key idea) **KYA.** Teeno tiles ko $h\to 0$ hone par compare karo. Strips **ek** direction mein shrink hoti hain ($\Delta u$ ya $\Delta v$ zero ki taraf jaata hai). Corner **dono** directions mein ek saath shrink hota hai — yeh do cheezein ka product hai jo zero ki taraf ja rahi hain, isliye yeh *bahut tezi se* shrink hota hai. **KYUN.** Sab kuch $h$ se divide karo (hum ek rate of change banana chahte hain, aur rate = change per unit $h$): $$ \frac{\Delta(uv)}{h} = v\,\underbrace{\frac{\Delta u}{h}}_{\to\, u'} + u\,\underbrace{\frac{\Delta v}{h}}_{\to\, v'} + \underbrace{\frac{\Delta u\,\Delta v}{h}}_{\to\, 0} $$ Last term dekho. Isko $\dfrac{\Delta u}{h}\cdot \Delta v$ likhao. Pehla factor finite number $u'$ ki taraf tend karta hai; doosra factor $\Delta v$ **$0$** ki taraf tend karta hai ($x$ ka ek baal jaisa push height mein ek baal jaisi movement laata hai). Finite $\times$ zero $=0$. **Corner mar jaata hai.** **PICTURE.** ![[deepdives/dd-maths-4.1.14-d2-s04.png]] Teen bars tiles ke sizes dikhate hain jab $h$ shrink hota hai (left se right): dono strips steady heights $v\,u'$ aur $u\,v'$ pe settle hoti hain, jabki corner bar zameen pe collapse ho jaata hai. > [!mistake] "Lekin humne corner ko $h$ se *divide* kiya — kya woh isse zinda nahi rakhta?" > **Kyun risky lagta hai:** ek tiny cheez ko tiny $h$ se divide karna kuch finite de sakta hai (jaise > $\Delta u/h \to u'$). > **Fix:** haan — lekin corner mein *do* tiny factors hain aur neeche sirf *ek* $h$ hai. $\Delta u/h$ apni tininess use karke $u'$ ban jaata hai, par phir bhi ek leftover $\Delta v \to 0$ bachta hai jise bachane ke liye koi $h$ nahi hai. Isliye corner truly vanish ho jaata hai. --- ## Step 5 — Wohi kaam limit definition se karo (algebra twin) **KYA.** Picture convincing hai, par aaiye isse [[Limit definition of the derivative]] se confirm karte hain. Definition ke mutabiq, $$ f'(x) = \lim_{h\to 0} \frac{u(x+h)\,v(x+h) - u(x)\,v(x)}{h} $$ Ab "strips mein kaatne" ka algebraic version: middle term $u(x+h)\,v(x)$ add aur subtract karo (net zero barabar hai, isliye legal hai): $$ u(x{+}h)v(x{+}h) - u(x)v(x) = \underbrace{u(x{+}h)\big[v(x{+}h)-v(x)\big]}_{\text{top strip}} + \underbrace{v(x)\big[u(x{+}h)-u(x)\big]}_{\text{right strip}} $$ Term by term: $u(x{+}h)$ **nayi width** hai; $[v(x{+}h)-v(x)]=\Delta v$ **extra height** hai — inка product top strip hai. Aur $v(x)$ **purani height** hai; $[u(x{+}h)-u(x)]=\Delta u$ **extra width** hai — inка product right strip hai. *Yeh Step 3 hai symbols mein likha gaya.* **KYUN.** Clever $\pm u(x+h)v(x)$ exactly woh tarika hai jisse hum ek $\Delta v$ aur ek $\Delta u$ force karte hain taaki har ek difference quotient ban sake. Corner kahan gaya? Woh $u(x{+}h)\,\Delta v$ ke andar chhupa hai — kyunki $u(x{+}h) = u + \Delta u$, us term mein secretly $u\,\Delta v$ (top strip) **aur** $\Delta u\,\Delta v$ (corner) dono hain ek saath. **PICTURE.** ![[deepdives/dd-maths-4.1.14-d2-s05.png]] Picture mein dono coloured brackets algebra terms se rectangle ke matching tiles tak point karte hain — symbols aur shapes ke beech ek dictionary. --- ## Step 6 — Piece by piece limit lo **KYA.** $h$ se divide karo aur $h\to 0$ hone do: $$ f'(x) = \lim_{h\to 0}\left[\, u(x{+}h)\,\frac{v(x{+}h)-v(x)}{h} \;+\; v(x)\,\frac{u(x{+}h)-u(x)}{h}\,\right] $$ Ab har limit padho: - $\dfrac{v(x{+}h)-v(x)}{h} \to v'(x)$ — yeh $v'$ ki definition hai. - $\dfrac{u(x{+}h)-u(x)}{h} \to u'(x)$ — yeh $u'$ ki definition hai. - $u(x{+}h) \to u(x)$ — **nayi width purani width pe wapas aa jaati hai** jab nudge vanish hoti hai. - $v(x)$ mein $h$ hai hi nahi, isliye woh waise hi baitha rehta hai. **KYUN us teesre limit ko care chahiye.** $u(x{+}h)\to u(x)$ kehna exactly woh statement hai ki $u$ **continuous** hai — koi achanak jump nahi. Hum yeh keh sakte hain *kyunki* $u$ differentiable hai, aur [[Differentiability implies continuity]]. Yeh woh quiet lemma hai jis par poora proof tika hai. **PICTURE.** ![[deepdives/dd-maths-4.1.14-d2-s06.png]] New-width marker (blue dot $u(x{+}h)$ par) ko old-width marker (yellow dot $u(x)$ par) ki taraf creep karte dikhaya gaya hai jab $h\to 0$ — continuity ek frame mein. Survivors ko assemble karo: $$ f'(x) = \underbrace{u(x)}_{\text{width wapas aayi}}\,\underbrace{v'(x)}_{\text{height ki speed}} + \underbrace{v(x)}_{\text{purani height}}\,\underbrace{u'(x)}_{\text{width ki speed}} \qquad\blacksquare $$ jise hum tidy karke headline banate hain: $(uv)' = u'v + uv'$. --- ## Step 7 — Edge cases: picture kabhi nahi toot ti Humne har scenario ka promise kiya tha. Rectangle argument lagta hai maanta hai ki $u,v>0$ hain aur dono badh rahe hain. Agar aisa na ho toh? **KYA / KYUN / PICTURE har case ke liye:** - **$u'=0$ (width frozen).** Right strip $v\,\Delta u$ ki zero width hai, isliye yeh kuch contribute nahi karta; sirf top strip bachti hai: $(uv)' = u\,v'$. Correct — constant width aur changing height sirf height ke through badhta hai. - **$v$ constant ($v'=0$).** Symmetric: $(uv)' = u'\,v$. Yeh **constant-multiple rule** hai, $(c\,u)' = c\,u'$, free mein nikal ke aayi. - **Ek side shrink hoti hai ($\Delta u<0$).** "Strip" add hone ki bajay daayein se *remove* hoti hai, isliye uska signed area $v\,\Delta u$ negative hai. Step 5 ki algebra ne kabhi sign assume nahi kiya — $\Delta u$ koi bhi real number ho sakta hai. Har conclusion unchanged rehta hai. - **Negative value ($u<0$ ya $v<0$).** Tab $uv$ ek *signed* area hai, par Steps 5–6 mein limit derivation real numbers par pure algebra hai aur usne kabhi positivity use nahi ki. Rectangle ek **thinking aid** hai; proof sab signs ke liye airtight hai. - **$h<0$ (peeche ki taraf nudge karna).** Limit $h\to 0$ dono sides include karta hai. Wohi teen quotients negative $h$ ke saath appear hote hain aur same $u',v'$ par converge karte hain. Koi case miss nahi hota. ![[deepdives/dd-maths-4.1.14-d2-s07.png]] Chaar mini-panels: frozen width, constant height, ek shrinking side (strip subtract hui), aur negative-valued (signed) rectangle — sab same rule pe land karte hain. --- ## Ek-picture summary ![[deepdives/dd-maths-4.1.14-d2-s08.png]] Ek frame mein poora argument hai: growing rectangle, uske teen tiles labelled $v\,\Delta u$, $u\,\Delta v$, aur doomed corner $\Delta u\,\Delta v$, ek arrow jis par "$\div h$, then $h\to 0$" likha hai, aur do surviving strips final line $(uv)' = u'v + uv'$ mein flow karti hain. > [!recall]- Feynman: poora walkthrough plain words mein batao > Ek rectangle imagine karo: itna wide, itna tall, area = width times height. Ab time ko ek whisker aage badhao. > Width thodi aur wide ho jaati hai aur height thodi aur tall — dono ek saath. Isse daayein ek thin strip add hoti hai (height × extra-width), upar ek thin strip (width × extra-height), aur ek tiny corner square jahan strips overlap karti hain. Rate paane ke liye, added area ko nudge ke size se divide karo aur nudge ko kuch nahi karo. Dono strips steady rates pe settle ho jaati hain — "height times how fast width grows" plus "width times how fast height grows." Corner mein do tiny factors the lekin divide karne ke liye sirf ek nudge tha, isliye woh bhookha mar jaata hai aur vanish ho jaata hai. Jo bachta hai woh product rule hai: $(uv)' = u'v + uv'$. Aur yeh parwah nahi karta ki sides badhein, ghate, freeze hon, ya negative hon — picture ke peeche ki algebra ne ek bhi sign use nahi kiya. > [!mnemonic] Ise chant karo: "**pehle ki speed** × doosra **plus** pehla × **doosre ki speed**." > $(uv)' = u'v + uv'$. --- ## Connections - [[Limit definition of the derivative]] — Steps 5–6 ka engine. - [[Differentiability implies continuity]] — Step 6 mein $u(x{+}h)\to u(x)$ kyun hota hai. - [[Power rule]] — $v'=0$ edge case aur parent ke induction se nikal ke aata hai. - [[Quotient rule]] — wohi rectangle idea ek division twist ke saath. - [[Chain rule]] — compositions ke liye uska partner. - [[Leibniz rule (nth derivative of a product)]] — yahi walkthrough, $n$ baar iterate kiya gaya. ## 🖼️ Concept Map ```mermaid flowchart TD R["Rectangle area A = u times v"] -->|"nudge x by h"| G["New area, both sides grow"] G --> RS["Right strip v times du"] G --> TS["Top strip u times dv"] G --> C["Corner du times dv"] RS -->|"divide by h"| L["Difference quotients"] TS -->|"divide by h"| L C -->|"two tinies, one h"| V["Corner vanishes"] L -->|"needs u continuous"| CO["Differentiable implies continuous"] L --> RULE["Product rule uv prime = u prime v + u v prime"] V --> RULE ```