Chain rule for multivariable functions — all cases

4.4.7 · Maths › Multivariable Calculus

Multivariable chain rule exist kyun karta hai?

Single-variable calculus mein, $y = f (x)$ , $x = g (t)$ se milta hai $\frac{d y}{d t} = \frac{d y}{d x} \frac{d x}{d t}$ — ek chain, ek path.

Lekin multivariable land mein, ek output $z = f (x, y)$ ke paas do doors hain ( $x$ aur $y$ ). Agar dono $x$ aur $y$ , $t$ pe depend karte hain, toh $t$ change karne se $z$ ek saath do doors se push hota hai. Hume dono pushes ka sum lena hoga.

First principles se derivation

Hum kahaan se shuru karte hain: $f$ ki differentiability ka matlab hai ek point ke paas, $Δ z = \frac{\partial f}{\partial x} Δ x + \frac{\partial f}{\partial y} Δ y + ε_{1} Δ x + ε_{2} Δ y,$ jahaan $ε_{1}, ε_{2} \to 0$ jab $(Δ x, Δ y) \to (0, 0)$ .

Rule kaise milta hai (case: $x = x (t), y = y (t)$ ): sab kuch $Δ t$ se divide karo: $\frac{Δ z}{Δ t} = \frac{\partial f}{\partial x} \frac{Δ x}{Δ t} + \frac{\partial f}{\partial y} \frac{Δ y}{Δ t} + ε_{1} \frac{Δ x}{Δ t} + ε_{2} \frac{Δ y}{Δ t} .$

Yeh step kyun? Hume $t$ ke respect mein ek rate chahiye, isliye har change ko per unit $Δ t$ measure karte hain.

$Δ t \to 0$ lo. Tab $Δ x, Δ y \to 0$ (continuity se), toh $ε_{1}, ε_{2} \to 0$ , aur last do terms vanish ho jaate hain:

Cases ka poora catalogue

Figure — Chain rule for multivariable functions — all cases

Case 2 — do independent variables: $z = f (x, y)$ , $x = x (s, t)$ , $y = y (s, t)$

Case 3 — general (woh formula jo sab ko contain karta hai)

Maano $u = f (x_{1}, \dots, x_{n})$ jahaan har $x_{i} = x_{i} (t_{1}, \dots, t_{m})$ .

Worked examples

Apni galtiyon ko steel-man karo

Recall Feynman: 12-saal ke bacche ko explain karo

Socho ek factory jo cookies banati hai (output). Cookies ki tadaad flour aur sugar pe depend karti hai. Lekin flour aur sugar dono ek hi farm se aate hain, aur farm ki harvest rain pe depend karti hai. Agar rain badhti hai, toh zyada flour AUR zyada sugar aata hai — dono cookie count badhate hain. Yeh dekhne ke liye ki rain cookies ko kitna change karti hai, tum dono supply roads follow karo (rain→flour→cookies, rain→sugar→cookies), raaste mein "har road kitna deliver karti hai" multiply karo, aur dono roads ke totals add karo. Yahi chain rule hai: deep cause se final result tak har road trace karo, ek road ke saath multiply karo, roads ke across add karo.

Connections

Partial Derivatives — building blocks $\partial z / \partial x$ .
Total Differential — woh linear approximation jise humne $d t$ se divide kiya.
Directional Derivative — polar form mein Case 2 ek directional derivative hai.
Gradient and Jacobian — general case = Jacobian matrix product.
Implicit Differentiation (multivariable) — explicit-term version use karta hai.
Single-variable Chain Rule — 1-path special case.

Flashcards

z = f (x, y)

ke liye multivariable chain rule jab

x = x (t), y = y (t)

\frac{d z}{d t} = \frac{\partial z}{\partial x} \frac{d x}{d t} + \frac{\partial z}{\partial y} \frac{d y}{d t}

Chain rule mein terms ADD kyun karte hain?

Kyunki total differential

d z = z_{x} d x + z_{y} d y

independent inputs ko independently (linearly) contribute karta hai, isliye effects add hote hain.

Chain rule mein

d

\partial

kab use karte hain?

d

tabhi jab function ultimately ek variable pe depend kare;

\partial

tabhi jab multiple ultimate variables baaki hon.

Path/tree method ek line mein

Har path ke saath derivatives multiply karo, phir output se variable tak ke sabhi paths ke upar add karo.

u = f (x_{1} .. x_{n})

x_{i} = x_{i} (t_{j})

ke liye general chain rule

\frac{\partial u}{\partial t _{j}} = \sum_{i = 1}^{n} \frac{\partial u}{\partial x _{i}} \frac{\partial x _{i}}{\partial t _{j}}

General chain rule ka matrix form

Jacobian product:

J_{u \circ x} = J_{u} J_{x}

w = f (x, y, t)

jab

x, y

t

pe depend karte hain:

\frac{d w}{d t} = ?

f_{x} x^{'} + f_{y} y^{'} + f_{t}

(explicit

t

term include karo).

x = r cos θ, y = r sin θ

ke liye

\partial z / \partial r

z_{x} cos θ + z_{y} sin θ