Foundations — Chain rule — proof, composite function derivatives

4.1.16 · D1 · Maths › Calculus I — Limits & Derivatives › Chain rule — proof, composite function derivatives

Parent note Chain Rule — Proof & Composite Function Derivatives padhne se pehle, aapko usmein aane wala har ek symbol achhe se samajhna hoga. Yeh page unhe ek-ek karke, bilkul scratch se, ek aisi order mein build karta hai jahan har nayi idea sirf pehle waali ideas pe depend karti hai.

1. Ek function — ek input aur ek output wali machine

Isko concrete banane wali picture: ek box jisme ek arrow andar jaata hai ($x$ label wala) aur ek arrow bahar aata hai ($f(x)$ label wala).

Letters $f$, $g$, $h$ sirf teen alag machines ke naam hain. Letters mein kuch special nahi — inhe Machine-1, Machine-2, Machine-3 bhi keh sakte the. Convention ke hisaab se:

• $g$ = inner machine (pehle chalti hai),
• $f$ = outer machine (baad mein chalti hai),
• $h$ = poori cheez jo ek combined machine ki tarah dekhi jaaye.

2. Composition $f(g(x))$ — do machines ko ek line mein wire karna

Figure ko left se right padho: $x$ andar $g$ mein jaata hai, ek middle number nikaltar aata hai, woh middle number $f$ mein jaata hai, final answer nikalkar aata hai. Middle number itna important hai ki agले section mein uska apna naam milta hai.

3. Letter $u$ — "middle number" ka naam

Upar wali figure mein, $u$ woh number hai jo middle arrow pe travel kar raha hai. Yeh koi naya mystery nahi — yeh literally $g(x)$ hai ek chhote se naam ke saath. Parent note likhta hai "with $u=g(x)$ and $y=f(u)$"; ab aap jaante ho $y$ sirf final output hai aur $u$ middle value hai.

4. $\Delta x$ — "input mein ek tiny nudge"

Number line pe $x$ pe bethe ek point ki picture karo; $\Delta x$ ek chota arrow hai jo use thoda right (ya left, agar $\Delta x$ negative ho) push kar raha hai.

Do aur deltas usi rule ko follow karte hain:

• $\Delta u=g(x+\Delta x)-g(x)$ — nudge ki wajah se middle number kitna hila,
• $\Delta y=f(u+\Delta u)-f(u)$ — final output kitna hila.

5. Ratio $\frac{\Delta y}{\Delta x}$ — "ek step pe steepness"

Figure dekho: curve pe do points lo, ek $x$ pe aur ek $x+\Delta x$ pe. Unhe milane wali straight line (secant) kheecho. Uski slope exactly $\frac{\Delta y}{\Delta x}$ hai — $\Delta x$ ke saath jao, $\Delta y$ upar jao.

6. Limit ${\lim }_{\Delta x\to 0}$ — "step ko kuch bhi nahi tak shrink karo"

Pichle figure ki secant line ki picture karo. Doosre point ko pehle ki taraf slide karo: $\Delta x$ shrink hota hai, secant pivot hoti hai, aur woh ek special line pe settle ho jaati hai — tangent, woh line jo curve ko us ek point pe graze karti hai. Limit wahi settling-down hai.

7. Derivative $f^{\prime }(x)$ — "ek point pe exact steepness"

Chota tick mark ${}^{\prime }$ ("prime") sirf matlab hai "iska derivative." Toh $g^{\prime }(x)$ inner machine ka stretch factor hai, aur $f^{\prime }(u)$ outer machine ka stretch factor hai middle value $u$ pe — dhyan se, $u$ pe evaluate hota hai, $x$ pe nahi.

8. Dono notations, side by side

Translation table:

• $\frac{dy}{du}$ = $f^{\prime }(u)$ = outer stretch factor,
• $\frac{du}{dx}$ = $g^{\prime }(x)$ = inner stretch factor,
• $\frac{dy}{dx}$ = $h^{\prime }(x)$ = total stretch factor.

9. Prerequisite map

Har cheez neeche chain rule ki taraf flow karti hai. Agar koi bhi upstream box shaky lage, uska section upar phir se padho.

Equipment checklist

Right side cover karo aur zor se bolke answer do — agar kar sako, toh parent note ke liye ready ho.