4.8.13
Ek derivative basically ek slope ki limit hai: f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x ) h → 0 h ruk jaate hain aur us resulting slope ko approximation ki tarah use karte hain.
Forward : dekhta hai function kahan ja raha hai (uses x x + h Backward : dekhta hai function kahan se aaya (uses x − h x Central : dono views ko average karta hai → yeh curvature ko symmetrically dekhta hai aur same h
WHAT kar rahe hain hum? f ′ ( x ) table of values ya ek aisi function se estimate karna jise hum sample kar sakein, bina exact symbolic derivative ke.
WHY matter karta hai yeh? Real problems mein f
HOW milte hain formulas? Taylor series se. Neeche jo bhi hai woh derive kiya gaya hai, kabhi seedha thopa nahi gaya.
x ± h
f ( x + h ) = f ( x ) + h f ′ ( x ) + 2 h 2 f ′′ ( x ) + 6 h 3 f ′′′ ( x ) + ⋯ f ( x − h ) = f ( x ) − h f ′ ( x ) + 2 h 2 f ′′ ( x ) − 6 h 3 f ′′′ ( x ) + ⋯
Yeh step kyun? Hum f ′ ( x ) f ′ ( x )
f ( x + h ) − f ( x ) = h f ′ ( x ) + 2 h 2 f ′′ ( x ) + ⋯ h f ′ ( x ) = h f ( x + h ) − f ( x ) − error 2 h f ′′ ( ξ )
Yeh step kyun? Same idea lekin f ( x − h )
f ( x ) − f ( x − h ) = h f ′ ( x ) − 2 h 2 f ′′ ( x ) + ⋯ f ′ ( x ) = h f ( x ) − f ( x − h ) + 2 h f ′′ ( ξ )
Yeh step kyun? Dono expansions ko subtract karo taaki symmetric f ′′
f ( x + h ) − f ( x − h ) = 2 h f ′ ( x ) + 6 2 h 3 f ′′′ ( x ) + ⋯ 2 h 2 f ′′ khatam ho jaate hain. 2 h
f ′ ( x ) = 2 h f ( x + h ) − f ( x − h ) − 6 h 2 f ′′′ ( ξ )
Yeh step kyun? Dono expansions ko add karo taaki odd terms (f ′ f ′′′ f ′′
f ( x + h ) + f ( x − h ) = 2 f ( x ) + h 2 f ′′ ( x ) + 12 h 4 f ′′′′ ( x ) + ⋯ f ′′
Worked example Example 1 — Teeno ko compare karof ( x ) = sin x x = 1 h = 0.1 f ′ ( 1 ) = cos 1 = 0.5403023 sin ( 0.9 ) = 0.7833269 , sin ( 1.0 ) = 0.8414710 , sin ( 1.1 ) = 0.8912074
Forward : 0.1 0.8912074 − 0.8414710 = 0.497364 0.0429 Yeh step kyun? Sirf x x + h ∼ 2 h ∣ f ′′ ∣ = 2 0.1 sin 1 ≈ 0.042
Backward : 0.1 0.8414710 − 0.7833269 = 0.581441 0.0411
Central : 0.2 0.8912074 − 0.7833269 = 0.539403 0.00090 Yeh step kyun? Forward aur backward ka average O ( h ) ∼ 6 h 2 ∣ f ′′′ ∣ = 6 0.01 cos 1 ≈ 0.0009
Lesson: same h
Worked example Example 2 —f ( x ) = e x x = 0 h = 0.1 f ′′ ( 0 ) = e 0 = 1 e 0.1 = 1.105171 , e 0 = 1 , e − 0.1 = 0.904837 f ′′ ( 0 ) ≈ 0.01 1.105171 − 2 ( 1 ) + 0.904837 = 0.01 0.010008 = 1.0008 Yeh step kyun? Combination f ( x + h ) − 2 f ( x ) + f ( x − h ) ∼ 12 h 2 f ′′′′ = 12 0.01 ≈ 0.0008
Worked example Example 3 — Ek raw table se (koi formula nahi pata)
x 2.0
2.1
2.2
f 0.6931
0.7419
0.7885
f ′ ( 2.1 ) h = 0.1 0.2 0.7885 − 0.6931 = 0.2 0.0954 = 0.477 Yeh step kyun? Hum central node 2.1 choose karte hain taaki dono neighbours use kar sakein. (Yeh data ln x f ′ ( 2.1 ) = 1/2.1 = 0.4762
h Kyun sahi lagta hai: math error → 0 h → 0 h Fix: numerator almost-equal numbers ka difference hai → catastrophic cancellation aur roundoff ε / h ≈ truncation ↓ 6 h 2 ∣ f ′′′ ∣ + roundoff ↑ h ε optimal h ε 1/3
Common mistake Central formula ko2 h h Kyun sahi lagta hai: forward/backward h h Fix: central x − h x + h 2 h answer aadha ho jaata hai.
Common mistake Interior point par forward difference use karna jab dono neighbours available hain.Kyun sahi lagta hai: forward pehla formula hota hai jo seekhte hain.
Fix: agar dono x ± h central ek order zyada accurate hai bilkul free mein. Forward/backward sirf table ke edges par use karo jahan ek neighbour missing ho.
Common mistake Error symbolO ( h ) O ( h 2 ) Kyun sahi lagta hai: dono h → 0 Fix: h 4 se. Kai steps mein yahi farq ek usable aur ek bekar answer ke beech hota hai.
Recall Woh 20% jo 80% deta hai
Central 1st: 2 h f ( x + h ) − f ( x − h ) O ( h 2 )
Central 2nd: h 2 f ( x + h ) − 2 f ( x ) + f ( x − h ) O ( h 2 )
Forward/backward O ( h )
Sab Taylor series se aate hain : subtract karo → odd derivative, add karo → even derivative.
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho tum ek pahaad par ho aur jaanna chahte ho ki bilkul wahan kitni dhaaldari hai jahan tum khade ho.
Forward : ek kadam aage lete ho, dekhte ho kitna upar/neeche gaye, yahi tumhara dhaaldari ka andaaza hai.Backward : same lekin ek kadam peeche lete ho.Central : ek kadam aage AND ek kadam peeche dono check karte ho, aur un dono jagahon ke beech ka farq use karte ho. Kyunki tumne dono taraf nazar daali, tumhara dhaaldari ka andaaza kaafi zyada sahi hota hai — chote-mote bumps cancel ho jaate hain.
Agar bahut bade kadam lo (h bahut hi chote kadam lo toh tumhara ruler itna precise nahi hota aur answer noisy ho jaata hai. Toh ek samajhdaari se chota step chuno.
Mnemonic Cancellation rule yaad rakhne ka tarika"SubtractOdd, AddEven" — Taylor expansions ko Sub tract karo taaki odd derivatives (f ′ Add karo taaki even derivatives (f ′′ C entral = C heta raho 2 h ka.
Forward difference formula for f ′ ( x ) h f ( x + h ) − f ( x ) O ( h )
Backward difference formula for f ′ ( x ) h f ( x ) − f ( x − h ) O ( h )
Central difference for f ′ ( x ) 2 h f ( x + h ) − f ( x − h ) O ( h 2 )
Central second-derivative formula? h 2 f ( x + h ) − 2 f ( x ) + f ( x − h ) O ( h 2 )
Central forward se zyada accurate kyun hai? Dono Taylor expansions subtract karne se f ′′ O ( h ) O ( h 2 )
Forward difference ka leading error term kya hai? − 2 h f ′′ ( ξ )
Central first-difference ka leading error term kya hai? − 6 h 2 f ′′′ ( ξ )
h Roundoff/cancellation error ε / h h
Even-derivative formula derive karne ki trick? f ( x + h ) f ( x − h )
Forward ya backward central ki jagah kab use karna chahiye? Data table ke edges par jab ek neighbour missing ho.
Finite step h approximation
Central second derivative
O of h squared second order
