Derivatives and rules (product, quotient, chain)

1.2.2 · AI-ML › Calculus & Optimization Basics

1. Derivative actually kya hota hai (first principles)

YEH definition KYUN? Slope = rise/run = $\frac{Δ y}{Δ x}$ . Ek finite interval $h$ par yeh sirf ek average slope hoti hai. Ek single point par slope paane ke liye hum run ko $h \to 0$ tak sikadte hain. Limit "ek window par average" ko "instantaneous rate" mein badal deti hai.

Power rule (general): $\frac{d}{d x} x^{n} = n x^{n - 1}$ . Yeh usi binomial expansion se aata hai — sirf $(1 n) x^{n - 1} h$ wala term $h$ se divide karne aur $h \to 0$ lene ke baad bachta hai.

2. Teeno combination rules

Tum rarely raw powers differentiate karte ho. Real functions build hoti hain multiply, divide, aur compose karke. Har rule ek construction ko handle karta hai.

2.1 Product rule

Derivation (WHY yeh do terms ka sum hai). $x$ ko $h$ se nudge karo. Product $uv$ se $(u + Δ u) (v + Δ v)$ ban jaata hai: $Δ (uv) = (u + Δ u) (v + Δ v) - uv = first order v Δ u + u Δ v + tiny Δ u Δ v$ $h$ se divide karo aur $h \to 0$ jaane do: $Δ u / h \to u^{'}$ , $Δ v / h \to v^{'}$ , aur $Δ u Δ v / h \to 0$ (do sikadte hue cheezon ka product). Isliye $u^{'} v + u v^{'}$ .

2.2 Quotient rule

Derivation. $q = u / v$ likho, toh $u = q v$ . Product rule se differentiate karo: $u^{'} = q^{'} v + q v^{'}$ . $q^{'}$ ke liye solve karo: $q^{'} = \frac{u ^{'} - q v ^{'}}{v} = \frac{u ^{'} - \frac{u}{v} v ^{'}}{v} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}$ Toh quotient rule nayi nahi hai — yeh product rule ulta solve karna hai.

2.3 Chain rule

Derivation (KYUN multiply karte hain?). Rates multiply hoti hain. Agar $u$ , $x$ se 3× tezi se badlta hai, aur $y$ , $u$ se 2× tezi se badlta hai, toh $y$ , $x$ se $2 \times 3 = 6$ × tezi se badlta hai: $\frac{Δ y}{Δ x} = \frac{Δ y}{Δ u} \cdot \frac{Δ u}{Δ x}$ $Δ u$ formally cancel ho jaata hai, aur $h \to 0$ lene se derivatives ka exact product milta hai.

Figure — Derivatives and rules (product, quotient, chain)

f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}

— tangent line ki slope.

Power rule

\frac{d}{d x} x^{n} = n x^{n - 1}

Product rule

(uv)^{'} = u^{'} v + u v^{'}

(do rectangle strips).

Quotient rule

(\frac{u}{v})^{'} = \frac{u ^{'} v - u v ^{'}}{v ^{2}}

Chain rule

f (g (x))

ke liye:

f^{'} (g (x)) \cdot g^{'} (x)

— rates (gear ratios) multiply hoti hain.

Product rule mein do terms kyun hoti hain?

u, v

sides wala rectangle dono taraf badhta hai; har side ek area strip add karti hai (

v Δ u

aur

u Δ v

Quotient rule kahan se aata hai?

u = q v

par product rule apply karo aur

q^{'}

ke liye solve karo.

Sigmoid

σ (x)

ki derivative

σ^{'} (x) = σ (x) (1 - σ (x))

Common product-rule error aur fix

(uv)^{'} \neq = u^{'} v^{'}

; sahi hai

u^{'} v + u v^{'}

Neural nets mein chain rule kisse correspond karta hai?

Backpropagation — gradients layer by layer multiply hote hain.

Recall Feynman: ek 12-saal ke bacche ko samjhao

Ek derivative sirf speed hai. Agar tumhari car ki position ek curve hai, toh derivative batata hai ki tum abhi kitni tezi se ja rahe ho.

Product rule: Socho ek photo jo ek saath wider BHI hoti hai aur taller BHI. Uska area DO wajahon se badhta hai — toh tum do growths ko add karte ho.
Chain rule: Gears! Agar ek chhota gear ek bada gear ghoomata hai jo ek bahut bada gear ghoomata hai, toh total spin saare gear ratios ka product hai.
Quotient rule: Pizza baantna. Agar pizza (top) badhta hai toh tumhe zyada milta hai; agar doston ki tadad (bottom) badhti hai toh tumhe kam milta hai — woh "kam" hi minus sign hai.

Connections

Limits and Continuity — derivative hi ek limit hai.
Gradient and Partial Derivatives — weight updates ke liye use hone wala multi-variable extension.
Backpropagation — network layers mein chain rule applied.
Gradient Descent — step direction decide karne ke liye derivatives use karta hai.
Activation Functions — inhi rules se sigmoid/tanh derivatives.
Power Rule and Basic Derivatives — atomic building blocks.

Derivatives and rules (product, quotient, chain)

1. Derivative actually kya hota hai (first principles)

2. Teeno combination rules

2.1 Product rule

2.2 Quotient rule

2.3 Chain rule

3. Worked examples

4. Common mistakes (steel-manned)

5. Flashcards

Connections

Concept Map