Implicit differentiation — technique, applications: Visual walkthrough

4.1.22 · D2 · Maths › Calculus I — Limits & Derivatives › Implicit differentiation — technique, applications

Step 0 — Woh words jo hum use karenge (scratch se banaye hue)

Koi bhi symbol aane se pehle, aao ek picture ko words mein fix karte hain.

Yahi puri cast hai. Neeche sab kuch $\frac{d y}{d x}$ dhundhne ke baare mein hai jab $y$ equation se bahar nahi aata.

Step 1 — Curve dekho aur woh number jo chahiye

KYA. Hum circle $x^{2} + y^{2} = 25$ draw karte hain aur uspar ek point choose karte hain, maano $P = (3, 4)$ (check karo: $9 + 16 = 25$ ✓). Hum $P$ par tangent line ka slope chahte hain.

KYUN. Pehle target dekhna algebra ko magic jaisi feeling se bachata hai. Slope ek real, visible cheez hai — $P$ par circle ko chhookar jaane wali red line ki tilt.

PICTURE. Figure mein red line circle ko $P$ par chhookar jaati hai aur paas mein kahin aur nahi. Yeh clearly downhill tilt karti hai (right move karne par neeche jaati hai), isliye hum pehle se hi expect karte hain ki $\frac{d y}{d x}$ is point par negative aayega.

Step 2 — Kyun hum pehle $y$ solve nahi kar sakte

KYA. $y$ ko free karne ki koshish karo: $x^{2} + y^{2} = 25$ se hume milta hai $y^{2} = 25 - x^{2}$ , isliye $y = \pm 25 - x^{2}$ .

KYUN. Woh $\pm$ hi poori problem hai. Ek equation secretly do functions rakhti hai — top half ( $+$ ) aur bottom half ( $-$ ). Ordinary tarike se differentiate karne ke liye hume ek branch choose karni padti aur square root ko chain rule se khiinchna padta. Gandha hai, aur doosri half ke liye phir se karna padta.

PICTURE. Circle un do red dots par split hoti hai jahan yeh vertical hai ( $x = \pm 5$ ). Dashed line ke upar $+$ branch hai; neeche $-$ branch. Koi ek formula $y = f (x)$ dono ko ek saath cover nahi karta.

Step 3 — Woh ek gear jo isse kaam karwata hai: toll

KYA. Hum Chain Rule ko uske sabse useful form mein yaad karte hain. Agar koi cheez $y$ se bani hai, aur $y$ khud $x$ ke saath chalti hai, toh

$\frac{d}{d x} (outer square y^{2}) = derivative of the square 2 y \cdot rate y itself moves \frac{d y}{d x} .$

KYUN. Jab hum $x$ ko thoda sa nudge karte hain, $x$ rate $1$ par change hota hai — isliye $\frac{d}{d x} (x^{2}) = 2 x \cdot 1 = 2 x$ , koi toll nahi. Lekin $y$ rate $1$ par change nahi hota; yeh rate $\frac{d y}{d x}$ par change hota hai (yeh equation ke through $x$ se tied hai). Chain rule har baar jab hum $y$ se guzarte hain toh woh rate ek "toll" ke roop mein charge karta hai.

PICTURE. Do side-by-side ramps. Left par, $x$ ko nudge karna $x^{2}$ ki height ko plain rate $2 x$ par move karta hai. Right par, $y^{2}$ change karne ke liye pehle $y$ ko move karna padta hai (rate $\frac{d y}{d x}$ , red arrow), phir square respond karta hai ( $2 y$ ). Red toll-arrow is pure page par akela naya idea hai.

Step 4 — Poori equation ko differentiate karo, term by term

KYA. $x^{2} + y^{2} = 25$ ke dono sides par $\frac{d}{d x}$ apply karo:

$= 2 x (toll-free) \frac{d}{d x} (x^{2}) + = 2 y \frac{d y}{d x} (pays toll) \frac{d}{d x} (y^{2}) = = 0 (constant, flat) \frac{d}{d x} (25)$

isliye

$from the x^{2} term 2 x + from the y^{2} term 2 y \frac{d y}{d x} = 25 never changes 0 .$

KYUN. Dono sides differentiate karna ek true equation ko true rakhta hai — agar do quantities hamesha equal hain, toh woh same rate par change hote hain. $25$ ek fixed number hai: yeh kabhi move nahi karta, isliye iska rate of change $0$ hai. $x^{2}$ term ordinary hai. $y^{2}$ term Step 3 se toll deta hai.

PICTURE. Equation ka har term uske "ek tiny push in $x$ par response" ke saath label kiya gaya hai: $x^{2}$ block $2 x$ se respond karta hai, $y^{2}$ block $2 y \cdot \frac{d y}{d x}$ se (red, toll ki wajah se), aur constant block nahi hilda.

Step 5 — Slope solve karo (Collect, Factor, Divide)

KYA. Ab hamare paas ek equation hai ek unknown ke saath, $\frac{d y}{d x}$ . Use isolate karo:

$2 y \frac{d y}{d x} = - 2 x ⟹ \frac{d y}{d x} = - \frac{2 x}{2 y} = - \frac{x}{y} .$

Symbol by symbol: $- 2 x$ woh $x^{2}$ term hai jo doosri side move ho gayi (Collect); $\frac{d y}{d x}$ pehle se akela khada hai isliye kuch Factor karne ki zaroorat nahi; $2 y$ se divide karna Divide hai; $2$ 's cancel ho jaate hain.

KYUN. Yeh ab plain algebra hai — calculus Step 4 mein khatam ho gaya. Dhyan do ki humein $y (x)$ ka formula kabhi nahi chahiya tha; answer mein $y$ mention ho sakta hai kyunki $y$ ek real number hai jab hum kisi point par khade hote hain.

PICTURE. $P = (3, 4)$ par formula deta hai $\frac{d y}{d x} = - \frac{3}{4} = - 0.75$ . Figure mein red tangent ki tilt exactly $3$ neeche hai har $4$ across ke liye — ek downhill line, jo hamare Step 1 ke prediction se match karta hai.

Step 6 — Har case: chaaon quadrants aur slope ka sign

KYA. Ek formula $- \frac{x}{y}$ circle ke chaaon regions ko silently handle karta hai. Har jagah uska sign padhte hain:

Quadrant	$x$ ka sign	$y$ ka sign	$- \frac{x}{y}$	tangent tilt
I (top-right)	$+$	$+$	$-$	downhill
II (top-left)	$-$	$+$	$+$	uphill
III (bottom-left)	$-$	$-$	$-$	downhill
IV (bottom-right)	$+$	$-$	$+$	uphill

KYUN. Ek explicit branch ke liye har half ke liye alag square-root formulas chahiye hote; yahan ek expression $y$ ke dono signs aur $x$ ke dono signs ko automatically cover karta hai. Yahi woh payoff hai jo parent note ne promise kiya tha: "dono branches ko ek saath cover karta hai."

PICTURE. Chaar sample points, ek ek quadrant mein, har ek apni red tangent ke saath. Dekho tilt exactly waisi flip karti hai uphill/downhill jaise sign table kehti hai.

Step 7 — Degenerate cases: jahan slope blow up ho jaata hai ya zero ho jaata hai

KYA. Do special situations:

Top aur bottom $(0, \pm 5)$ : yahan $x = 0$ , isliye $\frac{d y}{d x} = - \frac{0}{\pm 5} = 0$ . Flat tangent.
Left aur right $(\pm 5, 0)$ : yahan $y = 0$ , isliye $\frac{d y}{d x} = - \frac{\pm 5}{0}$ — undefined (division by zero). Vertical tangent.

KYUN. Formula randomly crash nahi karta; yeh exactly wahan fail karta hai jahan picture kehti hai slope ek number ke roop mein exist nahi karna chahiye. Ek vertical line ka "infinite" rise per run hota hai — koi finite slope nahi. Yahi exactly woh jagah bhi hai jahan $F_{y} = \frac{\partial}{\partial y} (x^{2} + y^{2}) = 2 y = 0$ hai, isliye Implicit Function Theorem humein warn karta hai ki $y$ wahan $x$ ka function nahi hai ( $\pm$ branches milti hain). Maths aur picture perfectly agree karte hain.

PICTURE. Red dots $(\pm 5, 0)$ par tangent ek vertical red line hai ( $y^{'}$ undefined); $(0, \pm 5)$ par tangent ek flat red line hai ( $y^{'} = 0$ ). Dono failure points Step 2 ke wahi do dots hain.

Ek-picture summary

KYA. Ek figure poora safar compress karti hai: equation, term-by-term toll, solved slope, aur four-quadrant sign pattern, sab ek hi circle par.

PICTURE. Circle apne chaar sample tangents carry karta hai (slope sign ke hisaab se colour-keyed), do degenerate red dots, aur beech mein boxed result $\frac{d y}{d x} = - \frac{x}{y}$ . Yeh akeli image hi derivation hai.

Recall Feynman: poora walkthrough plain words mein retell karo

Hum ek circle par kisi point ko barely chhookar jaane wali line ki tilt chahte the. $y$ solve karne se ek ugly $\pm$ split aayi, isliye humne solve karne se mana kar diya — humne bas promise kiya ki $y$ $x$ ka koi function hai aur poori equation differentiate kar di. Ek naya idea: jab bhi hum $y$ ko touch karein, hum toll dete hain, factor $\frac{d y}{d x}$ , kyunki $y$ $x$ ke saath chalti hai instead of still rehne ke. $x^{2} + y^{2} = 25$ ko term by term differentiate karne se mila $2 x + 2 y y^{'} = 0$ ; constant $25$ ne kuch contribute nahi kiya kyunki yeh kabhi nahi badlta. Thodi si algebra — collect, factor, divide — ne $y^{'} = - x / y$ ugal diya. Woh chhoti si formula sab kuch jaanti hai: top-right mein negative hai (downhill), top-left mein positive (uphill), bilkul top aur bottom par zero (flat), aur far left aur right par undefined (vertical) — exactly wahan jahan circle seedha khada hota hai aur koi slope number exist nahi kar sakta. Aur bonus mein usne ek geometry fact whisper kiya: tangent hamesha radius ke perpendicular hoti hai.

Connections

Implicit Differentiation — woh parent recipe jise yeh page draw karta hai.
Chain Rule — Step 3 mein "toll" gear exactly yahi hai.
Product Rule — jis moment $x y$ term appear ho tab chahiye (parent ke folium example mein dekho).
Tangent and Normal Lines — slope $- x / y$ kis kaam aata hai.
Implicit Function Theorem — " $y$ $x$ ka koi function hai" promise ko justify karta hai aur degenerate points predict karta hai.
Related Rates — same term-by-term toll, lekin hidden variable time $t$ hai.

Flashcards

x^{2} + y^{2} = 25

par,

\frac{d y}{d x}

kya hai?

- \frac{x}{y}

y^{2}

term "toll" kyun deta hai lekin

x^{2}

nahi deta?

y

rate

\frac{d y}{d x}

par change hota hai,

1

par nahi; chain rule woh factor attach karta hai.

x

rate

1

par change hota hai.

\frac{d}{d x} (25)

kitna hota hai aur kyun?

0

— ek constant kabhi nahi badlta.

Circle par tangent vertical kahan hai, aur formula wahan kya karta hai?

(\pm 5, 0)

par, jahan

y = 0

; formula division by zero deta hai (undefined slope).

Tangent flat kahan hai aur

y^{'}

wahan kya hai?

(0, \pm 5)

par, jahan

x = 0

;

y^{'} = 0

Quadrant II (top-left) mein

y^{'}

ka sign kya hai?

Positive (uphill):

x < 0

y > 0

isliye

- x / y > 0

y^{'} = - x / y

kaunsa geometric fact re-prove karta hai?

Tangent radius ke perpendicular hoti hai (unke slopes ka product

- 1

hota hai).