Foundations — Derivatives of all six trig functions

4.1.18 · D1 · Maths › Calculus I — Limits & Derivatives › Derivatives of all six trig functions

Is page par assume kiya gaya hai ki aapne kuch nahi dekha. Parent note the six trig derivatives ko touch karne se pehle, aapko har ek squiggle ke saath fluent hona chahiye jo woh likhta hai. Hum unhe ek ek karke build karte hain, har ek pichle ke upar.

1. Unit circle — woh stage jis par sab kuch baith ta hai

Trigonometry ki har cheez radius 1 ke ek circle par rehti hai, jo origin par centred hoti hai. Hum ise unit circle kehte hain kyunki "unit" ka matlab sirf "ek" hota hai.

Radians kyun aur degrees kyun nahi? Kyunki unit circle par angle arc length ke barabar sirf radians mein hota hai. Yeh akela fact hai jo baad mein $\frac{\sin h}{h}\to 1$ ko kaam karne deta hai. Degrees mein $\frac{\pi }{180}$ ka ek ugly factor aa jaata. Yeh woh "silent killer" hai jiske baare mein parent warn karta hai — aur yeh yahan shuru hota hai, definition mein.

2. $\sin$, $\cos$, aur circle par point

Ab point ko circle par khado aur use do axes par drop karo.

Yeh do numbers itni zyada kyun matter karti hain. Parent mein har derivative ek limit leta hai jab step $h\to 0$ hota hai, jo chote angle $h$ ko $0$ ki taraf drive karta hai. Us waqt $\cos h\to \cos 0=1$ aur $\sin h\to \sin 0=0$. Woh bilkul wahi values hain jo algebra ko collapse karti hain — inhe yaad rakho; yeh har trig limit ka landing spot hain.

Kyunki point radius 1 ke circle par baith ta hai, horizontal aur vertical legs aur radius ek right triangle banate hain jisme legs $\cos \theta$, $\sin \theta$ hain aur hypotenuse $1$ hai. Pythagoras tab hamen free mein identity deta hai: ${\sin }^2\theta +{\cos }^2\theta =1.$ Yahan ${\sin }^2\theta$, $(\sin \theta {)}^2$ ka shorthand hai — number $\sin \theta$ squared, na ki "sin of ${\theta }^2$". Yeh distinction rakho; parent ise constantly use karta hai.

2b. Negative angles — mirror symmetry

Kyunki limit $h\to 0$ mein $h$ zero se neeche se bhi aa sakta hai (ek choti negative nudge, ek clockwise rotation), hamen yeh jaanna chahiye ki $\sin$ aur $\cos$ negative angles ke liye kya karte hain.

Hamen iski zarurat kyun hai. ${\lim }_{h\to 0}$ mein step $h$ positive side aur negative side se approach karta hai. Ratio $\frac{\sin h}{h}$ ko dono se same value ki taraf jaana chahiye. Kyunki $\sin (-h)=-\sin h$ aur bottom bhi $-h$ ho jaata hai, do minus signs cancel ho jaate hain: $\frac{\sin (-h)}{-h}=\frac{\sin h}{h}$. Limit genuinely two-sided hai, aur yeh symmetry wajah hai ki dono sides agree karti hain.

3. Chaar ratio functions — $\tan ,\cot ,\sec ,\csc$

Baaki chaar trig functions nai geometry nahi hain. Woh $\sin$ aur $\cos$ ko ek doosre se divide karte hain. Isliye poora topic do facts par reduce hota hai.

4. Limit symbol $\underset{h\to 0}{\lim }$

Parent ka pehla move limit definition of the derivative hai. Us sentence ke sense banane se pehle aapko symbol padhna aana chahiye.

Hamen limit ki zarurat hi kyun hai. Slope ka matlab hai "rise over run", $\frac{\text{height mein change}}{\text{change in }x}$. Lekin instantaneous slope ke liye run zero hona chahiye — aur $\frac{0}{0}$ meaningless hai. Limit honest tarika hai yeh poochne ka ki "ratio kya hoga agar run kuch nahi ho sakta?" bina kabhi true zero se divide kiye. Yeh Limit definition of the derivative ka poora engine hai.

5. Derivative symbol $\frac{d}{dx}$

Parent mein har derivation bilkul is line ko likh kar shuru hota hai, phir $f$ ko $\sin$ ya $\cos$ se replace karta hai.

6. Angle addition — $\sin (x+h)$ ko expand kaise karein

Step $h$ angle ke andar add hota hai, $\sin (x+h)$ deta hai. Aage badne ke liye hamen ise alag karna hoga.

Yeh tool kyun aur doosra kyun nahi? Derivative hum par $x+h$ combination thopti hai. Fixed part $x$ ko shrinking part $h$ se alag karne ka sirf ek hi tarika hai — taaki limit akele $h$ par act kar sake — aur woh yeh identity hai. Yeh parent mein Steps 1 aur 2 ka hinge hai. Note karo: ek baar split hone par, $h$-pieces bilkul $\cos h$ aur $\sin h$ hain, jo $\cos 0=1$ aur $\sin 0=0$ ki taraf jaate hain — §2 se wahi values.

7. Woh rules jo sab kuch recombine karte hain

Jab ${\sin }^{\prime }$ aur ${\cos }^{\prime }$ maloom ho jaate hain, teen algebra rules kaam khatam karte hain. Aapko yahan sirf unki shapes pehchanni hain. Neeche har rule mein, $u$ aur $v$ sirf do choti functions ke naam hain jo aapne glue karke rakhi hain — is page par woh $\sin x$ aur $\cos x$ (ya constant $1$) ke liye khade honge. $u$ aur $v$ ko "top piece" aur "bottom piece" ki tarah sochho.

• Quotient rule workhorse hai, kyunki $\tan ,\cot ,\sec ,\csc$ mein se har ek $\sin$ aur $\cos$ ka fraction hai — toh $u$ aur $v$ $\sin x$ aur $\cos x$ ban jaate hain.
• Chain rule tab aata hai jab koi cheez trig function ke andar baith ti ho, jaise $\sec (3x)$ — inner $3x$ ko bhi differentiate karna hoga.

8. Sab kuch kaise fit hota hai

Circle hamen $\sin ,\cos$ aur unki starting values deta hai; woh do limits ko feed karte hain; limits ${\sin }^{\prime }$ aur ${\cos }^{\prime }$ deti hain; phir teen algebra rules un do facts ko sab chah functions tak spread karte hain.

Equipment checklist

Connections

• Derivatives of all six trig functions (woh parent jiske liye yeh page taiyaar karta hai)
• Limit definition of the derivative
• The Squeeze Theorem
• Angle addition formulas
• Pythagorean identity sin^2 + cos^2 = 1
• Quotient Rule
• Chain Rule
• Product Rule
• Simple Harmonic Motion
• Derivatives of inverse trig functions