Foundations — Chain rule for multivariable functions — all cases

4.4.7 · D1 · Maths › Multivariable Calculus › Chain rule for multivariable functions — all cases

Parent note padhne se pehle, tumhe ek chhota sa toolbox chahiye. Yeh page har woh symbol build karta hai jo woh use karta hai, bilkul scratch se shuru karke. Upar se neeche padho — har item apne pehle waale par lean karta hai.

0. Ek input ki function kya hoti hai? ($y=f(x)$)

Figure s01 — kya dekhna hai: literally ek box jisme baayi taraf $x$ label wala knob hai aur daayein taraf $y$ label wala dial. Input knob ghummao aur output dial swing karta hai. Box par naam $f$ likha hai — machine ka naam.

Letter $f$ bas machine ka naam hai. Agar hamare paas doosri machine hoti toh hum use $g$ keh sakte the. Parent note $f$ use karta hai main output machine ke liye aur $g$ input machine ke liye.

1. Rate of change: ordinary derivative $\frac{dy}{dx}$

Ab: jab tum input ko thoda sa nudge karo toh output kitni tezi se hilti hai? Yahi calculus ka poora point hai.

Figure s02 — kya dekhna hai: blue curve hai $y=f(x)$; orange straight line red point par bilkul touch karti hai. Chhota green right-triangle neeche tiny run $dx$ aur side par tiny rise $dy$ dikhata hai. Orange line ki slope $dy$ divided by $dx$ hai.

Yeh topic ko kyun chahiye. Chain rule rates of change ko "link by link" multiply karta hai. Har link exactly inhi ratios mein se ek hai. Agar tum $\frac{dy}{dx}$ ko "per unit input kitna output" nahi padh sakte, toh aage ka kuch bhi samajh nahi aayega.

Poore idea ka one-input version dekhne ke liye Single-variable Chain Rule dekho.

2. Do inputs ki functions: $z=f(x,y)$

Real machines mein aksar do knobs hote hain. Likho $z=f(x,y)$: do inputs $x$ aur $y$, ek output $z$.

Figure s03 — kya dekhna hai: flat curve ki jagah ab ek poori surface hover kar rahi hai. Floor par koi bhi point do numbers $(x,y)$ (east aur north) se name hota hai; uske seedha upar surface ki height $z$ hai. Red dot aur uski grey drop-line ek aisa point $(x,y,z)$ dikhate hain.

Yeh topic ko kyun chahiye. Parent note kehta hai ki ek output ke "do doors" hain $x$ aur $y$. Woh doors exactly do input slots hain. Ek wiggle kisi bhi door se aa sakti hai.

3. Partial derivative $\frac{\partial z}{\partial x}$ — slanted-d

Do knobs ke saath, "the" slope ambiguous hai — kis direction mein slope? Toh hum ek narrower sawaal poochhte hain: ek knob freeze karo, doosre ko wiggle karo.

Figure s04 — kya dekhna hai: $y$ ki ek fixed value chunte hain aur surface ko $x$-direction mein ek wall se slice karte hain. Orange curve us slice ka exposed edge hai — phir ek one-input curve. Is slice par red tangent line ki slope exactly $\frac{\partial z}{\partial x}$ hai. Is surface par marked point par slice crest ke baayein chadhti hai, toh dikhaya gaya slope positive hai ($+x$ direction mein uphill).

Yeh topic ko kyun chahiye. $z=f(x,y)$ se shuru hone wali multivariable chain mein har "link" ek partial derivative hai: $\frac{\partial z}{\partial x}$ aur $\frac{\partial z}{\partial y}$ do rates hain, ek per door.

Poori detail Partial Derivatives mein hai.

3b. $d$ kab use karein aur $\partial$ kab — reconciled

Yeh sabko trip karta hai, toh slowly padho. Choice depend karti hai upar wali cheez aakhirkar kitne variables par depend karti hai, poori chain mein neeche tak count karke.

4. Total differential $dz$ — dono doors combine karna

Ab dono knobs ko ek saath thoda sa nudge karo. $z$ mein total kitna change aata hai?

Figure s05 — kya dekhna hai: blue curved surface apne flat orange tangent plane ke saath red point par. Green step $dx$ east tumhe $z_{x}dx$ uthata hai; green step $dy$ north tumhe $z_{y}dy$ uthata hai; saath mein plane tumhe unke sum $dz$ se uthata hai. Step karne par plane aur true surface ke beech tiny gap woh error $\epsilon$ hai jo step se zyada tezi se vanish hoti hai.

Total differential se chain rule tak — limit written out. Maano do knobs ek single deeper variable $t$ se chalte hain (ek deep cause jaise time; uska tiny nudge $dt$ hai aur measurable nudge $\Delta t$ hai). Differentiability box se exact change se shuru karo aur har term ko measurable step $\Delta t$ se divide karo:

$\frac{\Delta z}{\Delta t}=\frac{\partial z}{\partial x}\frac{\Delta x}{\Delta t}+\frac{\partial z}{\partial y}\frac{\Delta y}{\Delta t}+{\epsilon }_1\frac{\Delta x}{\Delta t}+{\epsilon }_2\frac{\Delta y}{\Delta t}.$

Yeh legal kyun hai: ek sum ko $\Delta t$ se divide karna har piece ko alag-alag divide karta hai. Ab $\Delta t\to 0$ jaane do. Kyunki $x$ aur $y$ continuously $t$ ke saath chalte hain, steps $\Delta x,\Delta y\to 0$ bhi hote hain, jo ${\epsilon }_1,{\epsilon }_2\to 0$ force karta hai. Jabki ratios finite rehte hain: $\frac{\Delta x}{\Delta t}\to \frac{dx}{dt}$ aur $\frac{\Delta y}{\Delta t}\to \frac{dy}{dt}$. Toh last do terms hain (kuch $\to 0$) $\times$ (kuch finite) $\to 0$ aur bilkul drop ho jaate hain:

$\frac{dz}{dt}=\frac{\partial z}{\partial x}\frac{dx}{dt}+\frac{\partial z}{\partial y}\frac{dy}{dt}.$

Yahi ek deep variable ke liye poora chain rule hai, aur ab tumne error term ko faith par lene ki jagah dekha hai ki woh kaise marta hai. Note karo $z$ finally single variable $t$ par depend karta hai, toh straight $d$ $\frac{dz}{dt}$ mein correct hai (§3b). Total Differential dekho.

5. Composition: inputs jo deeper cheezein par depend karte hain

Yahan woh twist hai jo ise sirf partials ki jagah multivariable chain rule banata hai.

Dependency tree ek picture hai is "who-feeds-whom" ki: output upar baithta hai, aur arrows neeche point karte hain jis par bhi har cheez depend karti hai. Neeche ka map is poore page ke liye wahi tree hai — ise ek story ki tarah padho ki kaise har foundation doosri ko feed karti hai.

Map kaise padhen: top box y = f(x) se shuru karo — sabse simple machine. dy/dx ki taraf arrow kehta hai "ek input samajhne ke liye pehle uski rate of change chahiye". Woh two-input surface z = f(x,y) ko feed karta hai, jise partial derivative chahiye (ek knob freeze karo), jo hume total differential likhne deta hai (dono ko nudge karo, add karo). Use ek deep variable se divide karo aur tum composition tak pahuncho, aur finally Multivariable Chain Rule box tak. Do side-boxes — summation sign aur Jacobian — seedha us final box mein feed karte hain kyunki general chain rule dono chahiye. Koi bhi arrow follow karo aur padho "arrow ke peeche waale box se pehle arrow ke aage wala box chahiye".

Yeh topic ko kyun chahiye. Nesting ke bina koi chain follow karne ke liye nahi hai. Parent ki cookie story (rain → flour & sugar → cookies) exactly yahi hai: $t$ rain hai, $x,y$ flour aur sugar hain, $z$ cookies hain.

6. Summation sign $\sum$ aur index $i$

Jab bahut saare inputs $x_1,x_2,\dots ,x_n$ hote hain toh parent note sum ko ek symbol mein pack karta hai.

Yeh topic ko kyun chahiye. Jab $n$ doors hon toh har term likhna exhausting hai; ${\sum }_{i=1}^n$ same baat tiny space mein kehta hai, aur yahi exact form hai jo parent ka general rule use karta hai.

7. Gradient aur Jacobian — compact packaging

Partial derivatives ko ek single object mein collect karo taaki poora chain rule ek multiplication ban jaaye.

Parent ki "matrix form" remark ke liye tumhe bas inhein recognize karna hai. Deep dive: Gradient and Jacobian. Yeh Directional Derivative aur Implicit Differentiation (multivariable) connections ko bhi power karte hain jo parent list karta hai.

Equipment checklist

Khud test karo — daayein side cover karo. Agar koi answer blank hai, toh parent note se pehle woh section dobara padho.

Connections

• Single-variable Chain Rule — one-path warm-up.
• Partial Derivatives — chain ki har link inhi mein se ek hai.
• Total Differential — woh seed line jise hum $dt$ se divide karte hain.
• Directional Derivative — polar chain rule inhi mein se ek hai.
• Gradient and Jacobian — compact matrix packaging.
• Implicit Differentiation (multivariable) — explicit-term chain rule use karta hai.