Derivatives of sin x, cos x — proofs from first principles: Foundations

4.1.17 · D1 · Maths › Calculus I — Limits & Derivatives › Derivatives of sin x, cos x — proofs from first principles

Is page par kuch bhi assumed nahi hai. Har woh symbol jo parent note Derivatives of sin/cos use karta hai, yahan ground up se build kiya gaya hai, ek aisi order mein jahan har idea sirf usse pehle waale ideas par tikaa hai.

1. Angle — aur kyun hum ise radians mein measure karte hain

Hum is spread ko degrees (poora ghoomna = $360$ ) ya radians mein count kar sakte hain. Topic radians par hi insist karta hai, isliye hume samajhna chahiye ki radian kya hota hai.

Figure — Derivatives of sin x, cos x — proofs from first principles

Upar wala red arc dekho. Kyunki radius $1$ hai, poore circle ki circumference $2 π \cdot 1 = 2 π$ hai, toh ek poora ghoomna $2 π$ radians ka hota hai. Symbol $π$ (pi) bas yeh number hai $\approx 3.14159$ — kisi bhi circle ki circumference aur diameter ka ratio.

2. $sin$ aur $cos$ — circle par heights aur widths

Red vertical segment $sin x$ hai; black horizontal segment $cos x$ hai. Jaise tum $x$ ko sweep karte ho, $P$ circle trace karta hai, aur uski height $sin x$ wave ki tarah upar-neeche jaati hai.

Har quadrant mein signs — hume saare cases cover karne honge, kyunki $P$ chaaronon mein jaata hai:

Quadrant	angle range	$cos x$ (horizontal)	$sin x$ (vertical)
I	$0$ to $\frac{π}{2}$	$+$	$+$
II	$\frac{π}{2}$ to $π$	$-$	$+$
III	$π$ to $\frac{3 π}{2}$	$-$	$-$
IV	$\frac{3 π}{2}$ to $2 π$	$+$	$-$

3. Function symbol $f$ aur $f (x)$

Jab hum $f (x + h)$ likhte hain toh matlab hai: machine mein $x$ ki jagah number $x + h$ daalo. Toh $sin (x + h)$ thoda baad wale angle par ki height hai. Isse zyaada kuch nahi.

4. Limits — "dheeray se approach karo" wala symbol

Yeh kyun chahiye? Kyunki slope ke liye do points chahiye, lekin derivative chahti hai slope ek hi instant par. Hum do points lete hain $h$ distance par aur $h$ ko $0$ ki taraf shrink karte hain — lekin $h = 0$ set nahi kar sakte (woh $\frac{0}{0}$ deta hai, meaningless). Limit hi honest tarika hai yeh poochne ka ki "jaise gap khatam hoti hai kya hota hai?" Dekho Limits — definition and properties.

5. Derivative — ek point par curve ka slope

Do black dots hain $(x, f (x))$ aur $(x + h, f (x + h))$ . Rise hai $f (x + h) - f (x)$ , run hai $h$ , toh unhe join karne wali line (red line) ka slope hai $\frac{f ( x + h ) - f ( x )}{h} .$ Jaise $h$ shrink hota hai, woh red line true tangent ki taraf pivot karti hai. Limit lene se milta hai first-principles derivative: $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h} .$ Symbols: $f^{'} (x)$ ("f prime of x") aur $\frac{d}{d x}$ dono ka matlab hai "ka derivative." $\frac{d}{d x} sin x$ = "jis rate par $sin x$ change hota hai jaise $x$ change hota hai."

6. Addition formulas — algebraic splitter

Topic ko yeh kyun chahiye? Derivative machine mein $x + h$ daalta hai. Limit lene ke liye hume us part ko alag karna hoga jo $x$ par depend karta hai (jo fixed rehta hai) us part se jo $h$ par depend karta hai (jo shrink ho raha hai). Yahi formulas exactly woh surgery karte hain — woh $sin (x + h)$ ko ek $sin x$ chunk aur ek $cos x$ chunk mein chheel dete hain, har ek sirf $h$ waali cheez se multiply hota hai. Poori derivation yahan hai Trigonometric addition formulas.

7. Squeeze Theorem — ek limit ko trap karna

Socho ek coin do haathon ke beech dabaya hua hai jo paas aa rahe hain; coin ke paas kahi jaane ki jagah nahi siwaay wahan ke jahan haath milte hain. Parent $\frac{h}{s i n h}$ ko $1$ aur $\frac{1}{c o s h}$ ke beech teen circle areas use karke trap karta hai. Details yahan hain Squeeze Theorem.

Prerequisite map

Equipment checklist

Recall Apni readiness test karo (har answer reveal karo)

Ek radian kya hai, ek sentence mein? ::: Woh angle jiska arc unit circle par radius ke barabar length ka ho ( $= 1$ ). Unit circle par, $sin x$ point ki ___ hai aur $cos x$ ___ hai. ::: height (vertical position); width (horizontal position) Kin quadrants mein $sin x$ negative hota hai? ::: III aur IV (lower half). $f$ ka first-principles derivative likho. ::: $f^{'} (x) = lim_{h \to 0} \frac{f ( x + h ) - f ( x )}{h}$ Hum slope quotient mein simply $h = 0$ kyun nahi set kar sakte? ::: Woh $\frac{0}{0}$ deta hai, jo undefined hai — limit instead chupchaap paas jaati hai. Sine addition formula batao. ::: $sin (x + h) = sin x cos h + cos x sin h$ Cosine addition formula batao. ::: $cos (x + h) = cos x cos h - sin x sin h$ Squeeze Theorem se tum kya conclude kar sakte ho? ::: Agar $L \leq M \leq U$ aur dono $L, U \to$ same value, toh $M$ bhi wahan jaata hai. Calculus ke liye radians kyun, degrees kyun nahi? ::: Sirf radians mein hi ek tiny angle apne arc ke barabar hota hai, jo $\frac{s i n h}{h} \to 1$ ko clean banata hai. Kaunsi identity $cos^{2} h - 1$ ko $- sin^{2} h$ mein turn karti hai? ::: $cos^{2} h + sin^{2} h = 1$ (unit circle par Pythagoras).