Foundations — Quotient rule — proof

4.1.15 · D1 · Maths › Calculus I — Limits & Derivatives › Quotient rule — proof

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Parent proof bahut saara notation bahut tezi se throw karta hai: $u(x)$, $v(x)$, $f^{\prime }(x)$, ${\lim }_{h\to 0}$, $h$, difference quotients, "continuous", $v^2$. Agar inme se koi bhi fuzzy lag raha hai, to proof sirf symbol-pushing jaisi lagegi. Neeche hum har symbol ko ek ek karke earn karte hain, us order mein jis order mein proof unhe use karti hai, aur har ek ke saath ek picture bhi hai.

1. Ek function: ek aisi machine jo ek number leti hai aur ek number return karti hai

Picture. Ise ek vending machine ki tarah socho: button $x$ dabaao, snack $u(x)$ milta hai. Graph har (input, output) pair ko ek point $(x,u(x))$ ke roop mein draw karta hai, jo ek curve banata hai.

Cloze check: notation $v(x)$ ka matlab hai ==machine $v$ ka output jab input $x$ ho==.

2. Letter $h$: input ko ek chhoti si nudge

Picture. Horizontal axis par $x$ par khade ho jao. $h$ lambai ka ek chhota sa kadam right ki taraf lo; tum $x+h$ par pahunch jaate ho. Output $u(x)$ se badal kar $u(x+h)$ ho jaata hai.

3. Difference quotient: ek step par average steepness

Ratio kyun, aur yahi ratio kyun? Steepness sirf "output kitna badla" se nahi banti — lambe run par $2$ unit chadna gentle hai aur chhote run par steep. Steepness output-change per unit of input-change hai. Division exactly "per unit" ka sawaal answer karta hai, isliye hum divide karte hain.

4. Limit ${\lim }_{h\to 0}$: ratio kahan jakar settle hota hai

Picture. Jaise $h$ shrink hota hai, graph par do points slide hokar saath aa jaate hain. Unse guzarne wali seedhi line tilt hoti rehti hai jab tak woh tangent line nahi ban jaati — woh line jo curve ko $x$ par sirf graze karti hai. Uski slope woh value hai jis ki taraf difference quotient approach karta hai.

Poori engine ke liye dekho Limit definition of derivative.

5. Derivative $u^{\prime }(x)$: instantaneous rate of change

• $u^{\prime }$ bhi ek function hai: use $x$ do, woh wahan ki slope return karta hai.
• $u^{\prime }(x)>0$ matlab curve $x$ par rise kar raha hai; $u^{\prime }(x)<0$ matlab fall; $u^{\prime }(x)=0$ matlab flat.

6. Fraction $\frac{u}{v}$ aur uski denominator condition $v\ne 0$

$v$ ke sign ke saare cases. Rule tab bhi kaam karta hai jab $v(x)$ positive ya negative ho — sirf forbidden value exactly $0$ hai. Kyunki hum $v^2$ se divide karte hain (dekho §8), aur square kabhi negative nahi hota, formula ki shape dono taraf same rehti hai; sirf andar $v$ ki value change hoti hai.

7. Continuity: graph mein koi gap ya jump nahi

Picture. Ek continuous curve ek unbroken stroke hai. Ek discontinuous mein ek break hoti hai — break par, input ko thoda sa nudge karne par output bahut door jump kar sakta hai, isliye $v(x+h)$ $v(x)$ ki taraf approach nahi karta.

8. Powers aur notation $v^2$

Prerequisite map

Equipment checklist

Jab upar ke har reveal instant ho jaayein, tab tum Quotient rule — proof ke liye taiyaar ho.