Implicit differentiation — technique, applications

4.1.22 · Maths › Calculus I — Limits & Derivatives

WHAT hai implicit differentiation?

WHY kaam karta hai? (Derivation first principles se)

Poora mechanism ek assumption + ek rule pe tika hai.

Assumption. Kisi point ke paas, curve $F (x, y) = 0$ define karta hai $y$ ko kisi differentiable function $y (x)$ ke roop mein, chahe hum usse likh na sakein. (Implicit Function Theorem guarantee karta hai yeh jahan $\partial F / \partial y \neq = 0$ ho.)

Rule. Chain rule. Agar $u = g (y)$ aur $y = y (x)$ , toh $\frac{d}{d x} g (y) = g^{'} (y) \cdot \frac{d y}{d x} .$

Toh jab bhi tum kisi $y$ -wale term ko differentiate karte ho, ek extra factor $\frac{d y}{d x}$ nikal aata hai. Yahi ordinary differentiation se ek maatra farq hai.

General formula (derive karo, memorise mat karo)

$F (x, y) = 0$ ko $x$ ke w.r.t. multivariable chain rule se differentiate karo: $F_{x} + F_{y} \frac{d y}{d x} = 0 ⟹ \frac{d y}{d x} = - \frac{F _{x}}{F _{y}}$ jahan $F_{x} = \partial F / \partial x$ , $F_{y} = \partial F / \partial y$ . Yeh ek shortcut hai — lekin term-by-term method jo neeche hai, wahi actually use hota hai.

HOW — technique (4 steps)

Worked Examples

Applications

1. Tangent & normal lines un curves ke liye jo $y = f (x)$ form mein nahi hain

$(a, b)$ pe slope hai $m = \frac{d y}{d x}_{(a, b)}$ ; tangent $y - b = m (x - a)$ , normal ka slope $= - 1/ m$ .

2. Derivatives of inverse functions

3. Logarithmic differentiation (ek power tool)

Quantities jo ek equation se judi hain, dono time $t$ mein change ho rahi hain — $t$ ke w.r.t. implicitly differentiate karo. (Same chain-rule logic; $x$ aur $y$ dono $t$ ke functions hain.)

Second derivative implicitly

Common Mistakes (Steel-manned)

Recall Feynman: 12-year-old ko samjhao

Socho $x$ aur $y$ do dost hain jo ek rassi se bandhe hain (equation). Agar tum $x$ ko thoda hilao, $y$ ko bhi hilna padta hai — dono bandhe hain. Humein parwah nahi ki $y$ exactly kya hai; hum sirf poochhte hain "jab $x$ thoda sa move kare, $y$ kitna move karta hai?" Woh ratio hai $\frac{d y}{d x}$ . Har baar jab hum math mein kisi $y$ ko touch karte hain, hum multiply karte hain "y kitni tezi se move karta hai" se, kyunki $y$ $x$ ke saath dance kar raha hai, woh apni jagah khada nahi hai.

Flashcards

Implicit differentiation

y

ke baare mein kya assume karta hai?

y

x

ka (differentiable) function hai, chahe hum explicitly solve na kar sakein.

\frac{d}{d x} (y^{3})

ko extra factor kyun milta hai?

Chain rule —

y

andar hai, toh

\frac{d}{d x} (y^{3}) = 3 y^{2} \frac{d y}{d x}

\frac{d}{d x} (x y) = ?

y + x \frac{d y}{d x}

(product rule).

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

ke liye,

\frac{d y}{d x} = ?

- \frac{x}{y}

F (x, y) = 0

ke slope ka general formula?

\frac{d y}{d x} = - \frac{F _{x}}{F _{y}}

\frac{d}{d x} arcsin x

derive karne ka method

sin y = x

set karo, implicitly differentiate karo:

cos y y^{'} = 1

, toh

y^{'} = 1/ 1 - x^{2}

\frac{d}{d x} (x^{x}) = ?

aur technique?

Logarithmic differentiation;

x^{x} (ln x + 1)

Technique ke 4 steps?

Dono sides differentiate karo; har

y

-term mein

y^{'}

lagao;

y^{'}

terms collect karo; factor karo aur divide karo.

Implicit differentiation only option kab hota hai?

Jab equation ko

y

ke liye closed form mein solve nahi kar sakte (jaise

sin (x y) = x + y

Second-derivative ka trap?

y^{'}

ki expression wapas substitute karni hi padegi; final answer mein

y^{'}

nahi hona chahiye.

Connections

Chain Rule — har $\frac{d y}{d x}$ toll ke peeche ka engine.
Product Rule — mixed $x y$ terms ke liye zaroor hai.
Derivatives of Inverse Functions — implicit differentiation se derive hote hain.
Logarithmic Differentiation — variable bases/exponents.
Related Rates — time $t$ mein implicit differentiation.
Implicit Function Theorem — guarantee karta hai ki $y (x)$ exist karta hai jahan $F_{y} \neq = 0$ .
Tangent and Normal Lines — main geometric application.