Foundations — Partial derivatives — notation, calculation, geometric meaning

4.4.3 · D1 · Maths › Multivariable Calculus › Partial derivatives — notation, calculation, geometric meani

Is page mein kuch bhi assumed nahi hai. Agar parent note mein koi symbol likha tha, toh hum use yahan pehle build karte hain, order mein, har eent usse pehle waali eent par tikti hai.

1. Ek variable ka function — woh cheez jo aap pehle se jaante ho

Picture ek curve hai flat paper par. Horizontal axis input $x$ hai; vertical axis output $y=f(x)$ hai. Har input ek single height pin karta hai.

Yeh topic ko kyun chahiye: partial derivatives secretly ordinary one-variable derivatives hain. Agar aap $f(x)$ ko ek curve ki tarah picture nahi kar sakte, toh poora "freeze and differentiate" trick ka koi base nahi hai.

2. Slope — woh number jo derivative produce karta hai

$\text{slope}=\frac{\text{rise}}{\text{run}}=\frac{\text{change in output}}{\text{change in input}}$

Picture ek straight line hai jo curve ko ek point par sirf chhuuti hai — the tangent line. Uski steepness wahan ki slope hai.

Yeh topic ko kyun chahiye: partial derivative ko tangent line ki slope ke roop mein define kiya jaata hai. Slope woh currency hai jisme poora subject deal karta hai.

3. The limit ${\lim }_{h\to 0}$ — "step ko kuch bhi nahi hone do"

$h=0$ directly plug kyun nahi karte? Kyunki slope hai $\frac{\text{rise}}{\text{run}}$, aur $h=0$ par run $0$ ho jaata hai — zero se division, meaningless. Toh hum run ko $0$ ki taraf shrink karte hain aur dekhte hain ki ratio kya approach karta hai.

Yeh topic ko kyun chahiye: parent ki bilkul yahi definition hai $\frac{\partial f}{\partial x}(a,b)={\lim }_{h\to 0}\frac{f(a+h,b)-f(a,b)}{h}.$ Aap woh line tab tak nahi padh sakte jab tak yeh nahi jaante ki ${\lim }_{h\to 0}$ ka matlab hai "$h$ ko kuch nahi kar do."

4. Single-variable derivative $f^{\prime }(x)$

Teen rules jo aapko taiyaar rakhne chahiye (proved in Single-variable derivative):

Yeh topic ko kyun chahiye: poora computation method yahi hai — "doosra variable freeze karo, phir exactly yahi rules use karo." Koi naya calculus invent nahi hota — sirf reuse hota hai.

5. Ek saath do inputs — surface $z=f(x,y)$

Picture ab paper par curve nahi hai — ise 3D space chahiye. $x$ aur $y$ ko flat rakh do jaise ek map (the floor), aur $z$ ko floor ke upar ki height hone do. Floor par har point $(x,y)$ ko ek height milti hai, toh outputs ek surface banate hain — hills aur valleys ka ek landscape.

Yeh topic ko kyun chahiye: parent ki poori problem — "ek surface mein infinitely many directions hain, toh koi single slope nahi" — tab hi samajh mein aati hai jab aap $f(x,y)$ ko curve nahi, surface ki tarah dekho.

6. Point $(a,b)$ aur pair notation

Toh $f(a,b)$ woh single height hai jo us spot ke bilkul upar hai.

Yeh topic ko kyun chahiye: har evaluated partial ek point par likha jaata hai, jaise $f_x(1,2)=2$. Pair $(a,b)$ address hai; uske bina "slope yahan" ka koi yahan nahi hai.

7. Ek variable ko freeze karna — surface ko wapas curve banana

Yeh pivotal move hai, toh hum ise dheere-dheere build karte hain.

$y=b$ freeze karna aur sirf $x$ ko move hone dena ek brand-new one-variable function define karta hai: $g(x)=f(x,b).$ Iska graph exactly woh slice-curve hai — ek ordinary curve paper par phir se. Aur ab Sections 2–4 ki sab cheezein $g$ par apply hoti hain.

Yeh topic ko kyun chahiye: yahi wajah hai ki partial derivatives ordinary derivatives ke bhes mein hain. Parent ki key line $\frac{\partial f}{\partial x}(a,b)=g^{\prime }(a)$ bas yahi hai — "partial us sliced curve ka slope hai $x=a$ par."

8. Curly $\partial$ — ek nayi symbol ek nayi caution ke liye

Parent in sabko identical shorthands ke roop mein list karta hai: $\frac{\partial f}{\partial x}=f_x={\partial }_{x}f=D_{x}f.$ Chaalon ka ek hi number matlab hai. $f_x$ padhte hain "eff-sub-ex" — subscript us variable ka naam hai jo move hua.

Yeh topic ko kyun chahiye: parent ki har formula mein $\partial$ ya $f_x$ use hota hai. Is section mein woh glyphs apna matlab earn karte hain.

9. Stack kiye subscripts — higher aur mixed partials

Picture: $f_{xx}$ ek slice ki curvature hai (slope kaise muda hua hai); $f_{xy}$ surface ka twist hai — East slope kitna tilta hai jab aap North chalo. Woh twist symmetry hai Clairaut's theorem.

Yeh topic ko kyun chahiye: parent ka Clairaut section ($f_{xy}=f_{yx}$) tab tak unreadable hai jab tak yeh nahi pata ki stacked subscripts "phir se differentiate, is order mein" kaise encode karte hain.

10. Yeh kahaan jaata hai — tangent plane preview

Aapko yahan yeh derive nahi karna hai — woh hai Tangent plane and linear approximation mein. Bas ingredients pehchano: $f(a,b)$ height hai (Section 6), $f_x(a,b)$ aur $f_y(a,b)$ do slice-slopes hain (Sections 7–8), aur $(x-a),(y-b)$ yeh hain ki aap us spot se kitna door chale. Do directions mein do slopes ek flat sheet pin karte hain — the tangent plane.

Prerequisite map

Equipment checklist

Khud test karo — aage badhne se pehle har cheez ka jawab dena chahiye.

Connections

• Parent topic — woh note jise yeh foundations feed karte hain.
• Single-variable derivative — Sections 1–4 yahan poori tarah hain.
• Tangent plane and linear approximation — jahan do slopes combine hote hain.
• Gradient vector — $f_x,f_y$ ko saath package karta hai.
• Directional derivative — kisi bhi walking direction mein generalise karta hai.
• Clairaut's theorem — Section 9 ki twist symmetry.
• Chain rule (multivariable) · Total differential — agle steps.