Partial derivatives — Hinglish notes

1.2.3 · AI-ML › Calculus & Optimization Basics

WHY hum isko zaroori samjhein?

ML mein almost har loss function bahut saare parameters pe depend karta hai ek saath: $L (w_{1}, w_{2}, \dots, w_{n}, b)$ . Model train karne ke liye hum poochte hain "loss har weight ke liye alag-alag kaise respond karta hai?" taaki hum jaanein har weight ko kis direction mein push karein. Yeh per-variable slope hi partial derivative hai, aur inhe sab milake ek vector mein rakhne se gradient banta hai jis par gradient descent chalta hai.

HOW isko compute karein (mechanical rule)

WHAT karna hai: baaki har variable ko ek number maano aur normally differentiate karo.

WHY yeh kaam karta hai: upar wala limit sirf $x$ ko change karta hai; $y$ kabhi nahi hilta, toh derivative ke point of view se $y$ bilkul usi tarah behave karta hai jaise constant $5$ ya $π$ karta.

First principles se derivation

Maan lo $f (x, y) = x^{2} y + 3 y$ . $\frac{\partial f}{\partial x}$ sidha limit se compute karo.

$\frac{\partial f}{\partial x} = lim_{h \to 0} \frac{[ ( x + h ) ^{2} y + 3 y ] - [ x ^{2} y + 3 y ]}{h}$

Yeh step kyun? Hum $x + h$ ko har $x$ ki jagah plug karte hain, lekin $y$ ko waisa hi rehne dete hain (yahi definition hai).

$= lim_{h \to 0} \frac{( x ^{2} + 2 x h + h ^{2} ) y + 3 y - x ^{2} y - 3 y}{h}$

Yeh step kyun? $(x + h)^{2}$ ko expand karo taaki woh terms cancel ho jayein jo $h$ contain nahi karti.

$= lim_{h \to 0} \frac{2 x h y + h ^{2} y}{h} = lim_{h \to 0} (2 x y + h y) = 2 x y$

Yeh step kyun? $x^{2} y$ aur $3 y$ cancel ho jaate hain, hum $h$ factor karte hain, cancel karte hain, phir $h \to 0$ jaane dete hain. $3 y$ term gayab ho gayi kyunki usme koi $x$ nahi tha — $x$ -direction mein uska slope zero hai.

✅ Shortcut se same answer: $x^{2} y$ ko $2 x y$ differentiate karo ( $y$ constant ki tarah), aur $3 y$ ko $0$ .

Dual coding: surface $z = f (x, y)$ ko plane $y =$ const se slice karo. Us slice par ek 1-D curve milti hai; uska slope $\partial f / \partial x$ hai. $x =$ const se slice karne par $\partial f / \partial y$ milta hai.

Worked examples

Common mistakes

Flashcards

\partial f / \partial x

kya measure karta hai?

f

ka rate of change jab

x

change hota hai aur baaki sab variables constant hote hain.

\partial f / \partial x

compute karte waqt

y

ko kaise treat karte ho?

Ek constant ki tarah (jaise ek fixed number).

\partial f / \partial x

ki limit definition

lim_{h \to 0} \frac{f ( x + h , y ) - f ( x , y )}{h}

\partial (x^{2} y) / \partial x = ?

2 x y

\partial (e^{x y}) / \partial y = ?

x e^{x y}

(chain rule, inner factor

x

Gradient

\nabla f

kya hai?

Saare partial derivatives ka vector; steepest ascent ki direction mein point karta hai.

L = (w x + b - t)^{2}

ke liye,

\partial L / \partial w = ?

2 (w x + b - t) x

\partial / \partial x

mein koi

x

na wali term kyun vanish ho jaati hai?

x

-direction mein uska slope zero hai — jab sirf

x

move karta hai toh yeh change nahi hoti.

Recall Feynman: 12-saal ke bacche ko explain karo

Ek pahadi jagah imagine karo jahan tumhari height depend karti hai ki tum kitna east ( $x$ ) aur kitna north ( $y$ ) par khade ho. Partial derivative yeh hai: "Agar main sirf EK step EAST lun (north nahi), toh kya main upar jaata hun ya neeche, aur kitna steep hai?" Tum sirf east–west slope dekh ke jawab dete ho aur us waqt north–south tilt ko bilkul ignore karte ho. Phir north face karke wahi karo, aur ab tumhein dono directions mein slope pata hai — itna kaafi hai ki sabse tezi se neeche jaane ka raasta nikaal sako. Yahi "sabse tezi se neeche jaane ka raasta" exactly woh hai jo computer seekhne ke liye use karta hai.

Connections

1.2.02-Derivatives-single-variable — 1-D case jisme partial reduce hota hai.
1.2.04-Gradient-and-directional-derivatives — partials $\nabla f$ mein assembled.
1.2.05-Chain-rule-multivariable — backpropagation ke liye zaroori.
1.3.01-Gradient-descent — in partials ko use karke weights update karta hai.
Linear-regression — Example 3 uska exact learning rule hai.