Partial derivatives — notation, calculation, geometric meaning

4.4.3 · Maths › Multivariable Calculus

KYUN chahiye partial derivatives?

Single-variable calculus mein, $f (x)$ ka ek hi slope hota hai $f^{'} (x)$ . Lekin ek surface jaise $z = f (x, y)$ ke ek point par infinitely many directions hote hain jisme tum move kar sakte ho. Koi ek "slope" nahi hota. Isliye hum ek simpler, zyada honest sawaal poochte hain:

Agar main sirf $x$ -direction mein chalta hoon (jab $y$ fixed hai), toh height kitni tezi se change hogi?

Yeh deta hai partial derivative with respect to $x$ . Same idea $y$ ke liye. Yeh do special directions woh foundation hai jis par baaki sab kuch (gradient, directional derivative, tangent plane) build hota hai.

KYA hota hai partial derivative? (first principles se definition)

KYUN yeh ek ordinary derivative hi hai: ek single-variable function define karo $g (x) = f (x, b)$ ( $y = b$ freeze karo). Phir $\frac{\partial f}{\partial x} (a, b) = g^{'} (a) .$ Toh tumhare saare purane differentiation rules abhi bhi apply hote hain — bas frozen variable ko ek number ki tarah treat karo.

Notation (sabka matlab ek hi hai)

Ek point par evaluate karna: $\frac{\partial f}{\partial x}_{(a, b)}$ ya $f_{x} (a, b)$ .

KAISE calculate karein — freeze-and-differentiate algorithm

Woh variable chuno jiske respect mein differentiate karna hai.
Baaki har variable ko constant maano (ek fixed number jaise $7$ ).
Normally differentiate karo saare usual rules use karke (power, product, chain...).
Point plug in karo agar koi number maanga gaya ho.

GEOMETRIC meaning — surface ke slices

Figure — Partial derivatives — notation, calculation, geometric meaning

Yeh do slopes hi do tangent lines hain jo $(a, b)$ par tangent plane banate hain: $z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b) .$ KYUN: tangent plane ko dono slice-slopes reproduce karne chahiye, toh coefficients exactly partials hote hain.

Higher-order & mixed partials (ek jhaanki)

Tum phir se differentiate kar sakte ho: $f_{xx} = \partial_{x} (f_{x})$ , aur mixed $f_{x y} = \partial_{y} (f_{x})$ .

Common mistakes (steel-manned)

Active Recall

Recall Pehle forecast karo phir verify karo

Q: $f (x, y) = e^{x} cos y$ ke liye, padhne se pehle $f_{x}$ aur $f_{y}$ predict karo. A: $f_{x} = e^{x} cos y$ (cos y constant hai), $f_{y} = - e^{x} sin y$ ( $e^{x}$ constant hai).

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek pahadi maidan imagine karo. Tum ek jagah khade ho. Agar tum seedha East ki taraf EK kadam lete ho aur dekhte ho ki tum kitna upar ya neeche gaye, toh woh "East slope" hai — $x$ mein partial derivative. Ek kadam North ki taraf lete ho → woh "North slope" hai, $y$ mein partial. Tum dono directions ko alag-alag measure karte ho, yeh maanke ki tum sirf us ek direction mein chal sakte ho. Pahaad nahi badalta; tum bas choose karte ho ki kaunsa rasta test karna hai.

Flashcards

\partial f / \partial x

kya measure karta hai?

f

ke change ki rate jab

x

vary karta hai aur baaki saari variables constant hain.

f_{x} (a, b)

ki limit definition?

lim_{h \to 0} \frac{f ( a + h , b ) - f ( a , b )}{h}

Practically partial derivative kaise compute karte hain?

Baaki saari variables ko constants maano aur normally differentiate karo.

f_{x} (a, b)

ka geometric meaning?

Plane

y = b

se kati curve ki tangent line ka slope

x = a

par.

d

ki jagah

\partial

kyun use karte hain?

Kyunki

f

kai variables par depend karta hai;

\partial

signal karta hai ki doosre fixed hain.

f = x^{2} y

f_{x}

nikalo.

2 x y

(

2 x

nahi —

y

ko rakhna hai).

f = sin (x y^{2})

f_{y}

nikalo.

cos (x y^{2}) \cdot 2 x y

Clairaut's theorem kya kehta hai?

Agar second partials continuous hain, toh

f_{x y} = f_{y x}

(a, b)

par tangent plane formula?

z = f (a, b) + f_{x} (a, b) (x - a) + f_{y} (a, b) (y - b)

Common error:

f_{x}

mein

- 7 y

differentiate karna?

Yeh

x

ke respect mein constant hai, toh iski

x

-derivative

0

hai.

Connections

Gradient vector — $f_{x}, f_{y}$ ko $\nabla f$ mein package karta hai.
Directional derivative — partials ko kisi bhi direction mein generalize karta hai.
Tangent plane and linear approximation — dono partials se bana hai.
Chain rule (multivariable) — partials chain hote hain.
Total differential — $df = f_{x} d x + f_{y} d y$ .
Single-variable derivative — woh special case jis mein partials reduce ho jaate hain.
Clairaut's theorem — mixed partials ki symmetry.