4.1.10 · D1Derivative from first principles — difference quotient defin
Intuition Ek hi core ideaEk derivative hoti hai ek single point par curve ki steepness , jo do points ke beech slope measure karke aur phir unhe ek saath slide karke milate hue nikali jaati hai. Neeche jo kuch bhi hai woh wahi vocabulary aur pictures hain jo tumhe chahiye taaki jab parent note lim h → 0 h f ( x + h ) − f ( x )
Is page par assume kiya gaya hai ki tum kuch nahi jaante "ek graph numbers ki picture hoti hai" se aage. Hum har ek letter khud kamaenge, ek ek karke, aur har naya idea pichle idea par tikaa hoga.
f f machine hai: tum isme ek number daalte ho, woh ek number wapas deti hai. Hum f ( x ) x x input slot hai.
f x f ( x ) output ).
Ek vending machine socho. Button x = 3 f ( 3 ) = 9 f ( x ) = x 2 Rule fixed hai; sirf tumhara input badalta hai.
Topic ko yeh kyun chahiye: derivative ek function ke baare mein ek fact hai. Koi machine nahi, differentiate karne ko kuch nahi.
f x f ( x ) Point ( x , f ( x ) ) x f ( x ) x curve ban jaate hain. Wahi curve graph hai.
Horizontal axis input x Vertical axis output f ( x ) Curve par ek single point ( x , f ( x ) )
Intuition Ek line kyun nahi, curve kyun?Agar bade inputs proportionally bade outputs dete hain, toh straight line milti hai. Lekin f ( x ) = x 2 tezi se aur tezi se badtha hai, isliye picture upar ki taraf bend hoti hai — ek curve. Us curve ki alag alag jagahon par alag alag tilt hoti hai, aur wahi tilt is puri topic ki kahaani hai .
Topic ko yeh kyun chahiye: "slope" aur "steepness" woh cheezein hain jo tum graph par dekhte ho. Picture ke bina steepness measure karne ko kuch nahi hai.
Definition Ek straight line ki slopeSlope yeh bataati hai: "har ek sideways step ke liye, main kitna upar chadtha hoon?" Line par do points lo. Run hai kitna aage gaye; rise hai kitna upar chadhе. Tab
slope = run rise = change in sideways change in height
Run = input mein change (horizontal).Rise = output mein change (vertical).Badi slope = steep. 0 Negative slope = aage move karne par neeche jaana.
Intuition "Over" kyun? Divide kyun karte hain?Rise ko run se divide karne par har ek sideways step par chadhai milti hai. Isse slopes comparable ban jaati hain: ek staircase jo 4 m mein 2 m chadhti hai (2 1 3
Common mistake Slope ke liye DO points chahiyeEk single dot se rise-ya-run compute nahi ho sakta — change ke liye ek "pehle" aur ek "baad" chahiye. Yeh akela fact hi is puri topic ka reason hai: ek curve ki slope har point par badalty hai, phir bhi slope ko do points chahiye lagte hain. Woh trick (Section 6) ise resolve karti hai.
Topic ko yeh kyun chahiye: derivative ek slope hi hai . Parent ke har worked example ka end ek slope value par hota hai.
Average vs Instantaneous Rate of Change dekho — do-point slope ek average rate hai; derivative instantaneous wali hoti hai.
h h chhoti si distance hai jitni hum input axis par right taraf kadam bhadte hain . Input x x + h h 0.1 0.001
x x + h buddy point .h run ).
Intuition Buddy ko picture karoApni ungli curve par x x + h h
h h zinda hai (non-zero) jab tak hum algebra karte hain. Sirf aakhri step par hum poochhte hain ki jab yeh zero ki taraf jaata hai tab kya hota hai. h = 0
Topic ko yeh kyun chahiye: h
f ( x + h ) x + h f ( x + h ) buddy point par curve ki height hai. Yeh milta hai rule mein har jagah x ( x + h )
Worked example Concrete substitutionAgar f ( x ) = x 2 f ( x + h ) = ( x + h ) 2 ( x + h ) 2 = x 2 + 2 x h + h 2 f ( x ) = x 1 f ( x + h ) = x + h 1
Common mistake HAR JAGAH substitute karof ( x ) = x 2 ( x + h ) 2 na ki x 2 + h x + h
Topic ko yeh kyun chahiye: slope measure karne ke liye tumhe dono points par height chahiye — f ( x ) f ( x + h )
Ab hum Sections 3, 4, 5 combine karte hain. Do points hain:
Point A: address ( x , f ( x ) )
Point B: address ( x + h , f ( x + h ) )
Rise (A se B tak chadhai) hai f ( x + h ) − f ( x ) Run (A se B tak sideways) hai ( x + h ) − x = h
Secant woh koi bhi straight line hoti hai jo ek curve ko do points par kaatti hai. Iska slope un do points ke beech curve ki average steepness hai.
Topic ko yeh kyun chahiye: yahi woh object hai jis par puri topic bani hai. Derivative woh hai jis par yeh quantity approach karti hai jab buddy slide karke aata hai. Secant and Tangent lines dekho.
lim h → 0 ( ⋯ ) "woh value jis par expression settle hoti hai jab h 0 0 Arrow → destination ki parwah karte hain, endpoint ki nahi.
Intuition Yahan limit kyun chahiyeJaise jaise buddy point tumhare point ki taraf slide karta hai (h → 0 rotate hoti hai aur, merged position mein, curve ko sirf touch karti hai — woh touching line tangent hai. Lekin hum exactly h = 0 kahan ja rahi hai . Limit woh tool hai jo "kahan ja rahi hai?" ka jawab deta hai — exactly woh sawaal jo yeh topic poochhta hai aur koi doosra tool nahi deta. Limits — formal definition and one-sided limits dekho.
Tangent curve ko ek point par touch karti hai aur wahin uski steepness se match karti hai. Iska slope derivative hai . Jaise h → 0 →
Topic ko yeh kyun chahiye: limit ke bina "ek single point par slope" impossible hai — run 0
Agar hum simplify karne se pehle h = 0 0 f ( x ) − f ( x ) = 0 0 0
0 0 0 0 0 indeterminate form kehte hain — kuch aur kaam karne tak undecided. Indeterminate forms 0 over 0 dekho.
Pehle algebra se troublesome h cancel karo (upar se factor out karo, neeche se cancel karo). Cancel hone ke baad expression safe hai aur tum h → 0
Topic ko yeh kyun chahiye: yeh strict order explain karta hai — pehle simplify, phir limit lo — jo poore method ko kaam karata hai.
f ′ ( x ) ′ prime ) ka matlab hai "ki derivative." Toh f ′ ( x ) f x point x f . Yeh khud ek nayi machine hai: ek point input karo, wahin steepness output milti hai.
f ′ f f ( 10 ) = 100 f ′ ( 10 ) = 20 f ( x ) = x 2 height hai, doosra steepness hai. Kabhi f ′ f f Continuity and Differentiability se related hai.
Topic ko yeh kyun chahiye: f ′ ( x ) answer hai jo topic produce karta hai — position ki function ke roop mein derivative.
Ab parent note ka headline koi ajnabi nahi chhodta:
f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x )
Ise ek sentence ki tarah padho: "x f f ′ ( x ) the fraction ) jaati hai (lim h → 0
Graph turns machine into a curve
h is a tiny step to x plus h
f of x plus h is height at buddy
Difference quotient secant slope
Zero over zero is forbidden
Khud test karo — right side cover karo aur zor se jawab do.
f ( x ) Woh number jo ek machine f x
Graph par, point ( x , f ( x ) ) x f ( x )
Bina symbols ke slope define karo. Rise over run — har ek sideways step ke liye kitna chadthe ho.
h Buddy point x + h non-zero hai.
f ( x ) = x 2 f ( x + h ) Har x ( x + h ) ( x + h ) 2 = x 2 + 2 x h + h 2
h f ( x + h ) − f ( x ) Do curve points se guzarne wali secant line ki slope.
lim h → 0 Woh value jo expression settle karti hai jab h 0
Turant h = 0 Run 0 0 0
f ′ ( x ) Point x f
Jaise h → 0 tangent
