4.1.17
Intuition Badi picture (KYUN)Sine wave sabse tezi se rise karta hai jab wo zero cross karta hai, aur apni peaks par flat ho jaata hai. Yeh "rate of climb" khud ek wave hai — aur yeh nikalta hai cosine wave (jo exactly wahan peak karta hai jahan sine upar jaate hue zero cross karta hai). Toh differentiation bas wave ko 90° shift kar deta hai. Hum ise derivative ki raw definition se prove karne waale hain, memorize nahi karna.
Definition Derivative (first principles)Kisi function f f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x ) d x d sin x === cos x == , d x d cos x === − sin x == (angles radians mein — yeh zaroori hai).
Sab kuch do limits par depend karta hai. Inke bina proof collapse ho jaata hai.
Worked example Poori derivation — har step justifiedd x d sin x = lim h → 0 h s i n ( x + h ) − s i n x Step 1 — addition formula se expand karo. Kyun? Hume "x h sin ( x + h ) = sin x cos h + cos x sin h = lim h → 0 h s i n x c o s h + c o s x s i n h − s i n x Step 2 — sin x Kyun? Do jaane-maane limits build karne ke liye.
= lim h → 0 [ sin x ⋅ h c o s h − 1 + cos x ⋅ h s i n h ] Step 3 — sin x , cos x h , toh inhe bahar nikalo:
= sin x = 0 h → 0 lim h cos h − 1 + cos x = 1 h → 0 lim h sin h Step 4 — do ingredients substitute karo:
= sin x ⋅ 0 + cos x ⋅ 1 = cos x ✓
Worked example Poori derivation — upar wale ka mirrord x d cos x = lim h → 0 h c o s ( x + h ) − c o s x Step 1 — addition formula cos ( x + h ) = cos x cos h − sin x sin h Kyun? Same reason: x h = lim h → 0 h c o s x c o s h − s i n x s i n h − c o s x Step 2 — cos x
= lim h → 0 [ cos x ⋅ h c o s h − 1 − sin x ⋅ h s i n h ] Step 3 — constants bahar nikalo aur limits substitute karo (0 1
= cos x ⋅ 0 − sin x ⋅ 1 = − sin x ✓
Intuition Sign ka sanity checkx = 0 cos x peak par hai aur decrease hona shuru karta hai → slope negative. Aur sach mein − sin x x > 0 physically forced hai. ✔
Recall Dekhne se pehle predict karoQ: Bina re-derive kiye, first principles structure se d x d sin ( 2 x ) sin ( 2 x + 2 h ) A: Wohi proof h → 2 h 2 cos ( 2 x ) h s i n 2 h → 2 chain rule ka preview hai.
Common mistake "Degrees mein kaam karna theek hai."Kyun sahi lagta hai: Degrees angles ki roz ki unit hain, toh kyun nahi?
Fix: Limit 1, h s i n h → 1 sirf radians mein sach hai (sector area 2 1 h h s i n h ° → 180 π d x d sin x = 180 π cos x Poore calculus mein radians use hote hain.
lim h c o s h − 1 = 2 1 1 − 2 h 2 Kyun sahi lagta hai: cos h ≈ 1 − 2 h 2 h c o s h − 1 ≈ − 2 h Fix: Woh − 2 h → 0 2 1 h 2 1 − c o s h → 2 1 alag limit hai. Denominator mein h
h s i n h Kyun sahi lagta hai: Yeh 0 0 Fix: L'Hôpital ko d h d sin h = cos h Circular reasoning. Squeeze argument hi legitimate foundation hai.
First-principles definition of f ′ ( x ) h → 0 lim h f ( x + h ) − f ( x )
lim h → 0 h s i n h 1
lim h → 0 h c o s h − 1 0
Kaun sa theorem h s i n h → 1 Squeeze (Sandwich) Theorem, unit-circle areas ke through
Squeeze mein teen areas kaun se compare hote hain? 2 1 sin h ≤ 2 1 h ≤ 2 1 tan h
sin sin ( x + h ) = sin x cos h + cos x sin h
cos cos ( x + h ) = cos x cos h − sin x sin h
d x d sin x cos x
d x d cos x − sin x
Angles radians mein kyun hone chahiye? Kyunki h s i n h → 1 = 2 1 h
h c o s h − 1 Conjugate c o s h + 1 c o s h + 1
Yahan L'Hôpital kyun use nahi kar sakte? Yeh d x d sin x = cos x
Recall Feynman: 12-saal ke bacche ko samjhaoSocho kisi swing ko dhakka de rahe ho. Jab swing bilkul neeche hoti hai toh sabse tezi se move karti hai; upar jaake ek pal ke liye ruk jaati hai. Swing ki height sin x kitni tezi se move kar rahi hai woh cos x prove karne ke liye hum super close zoom karte hain (yahi limit hai), ek circle use karke dikhate hain ki ek chhota angle aur uska sine almost equal hote hain, aur algebra apni jagah fit ho jaati hai.
Mnemonic Signs yaad rakhne ka tarika"Sine se Cosine free mein milta hai; Cosine se Sine ke liye fee (minus) lagti hai."
Ya cycle: sin → cos → − sin → − cos → sin 90°
Agar sirf ek cheez yaad rakhni ho: poora result h s i n h → 1 Inhe master karo aur tum dono derivatives 60 seconds mein rebuild kar sakte ho.
Squeeze Theorem — h s i n h → 1 Trigonometric addition formulas — algebraic splitterLimits — definition and properties Chain Rule — sin ( g ( x )) Derivatives of tan, sec, csc, cot — inhi do se quotient rule ke through bante hainTaylor series of sin and cos — usi fact ka alternative viewpointRadian measure — kyun constant clean aata hai
First principles derivative limit
Limit sin h over h equals 1
Limit cos h minus 1 over h equals 0
Squeeze Theorem on unit circle areas
Conjugate trick times cos h plus 1