4.1.12 · D1Power rule — proof for integer, rational exponents
Poora power rule basically ek seedha sawal hai — "jab main input x x n — aur iska jawab careful algebra se diya gaya hai. Neeche har ek symbol sirf usi nudge ko banana, movement ko measure karna, aur nudge ko zero tak cleanly shrink karne ke liye exist karta hai.
Parent proof ko padhne se pehle, tumhe page par har ek mark ko apna banana hoga. Neeche, har symbol ko zero se build kiya gaya hai: plain-words meaning, ek picture, aur woh reason ki proof uske bina ji nahi sakti. Inhe order mein padho — har ek pichle wale par lean karta hai.
x x a number we are free to change hai. Ek ruler par slider imagine karo; jahan bhi park karo, wahi current x jab yeh slider thoda sa move kare .
n x n x n x 3 = x ⋅ x ⋅ x n exponent (ya power ) hai; bada number x base hai.
Parent note ka "tower of blocks" exactly yahi hai: n x height n volume x n
Intuition Proof specificallyx n x 2 x 3 x 1/2 x − 1 x x n har tarah ke n ke liye differentiate kar sakein, toh humne ek saath lagbhag har derivative unlock kar li.
Common mistake Base vs. exponent — inhe swap mat karox n variable base x constant exponent n n x 2 x pehle wale ke baare mein hai. Doosre ke liye Derivative of exponential functions dekho.
Parent proof stages mein n
Definition Exponents ki families
Positive integers { 1 , 2 , 3 , … } Z + Zero , n = 0 koi multiplication nahi — convention se x 0 = 1 Negative integers { − 1 , − 2 , − 3 , … } reciprocal imagine karo: x − m = x m 1 Rationals q p Q Roots imagine karo: x 1/2 = x x 2/3 = 3 x 2
Intuition Inn exact stages mein kyun split kiya?Kyunki har type ko ek alag tool chahiye (tum x 1/2 x 3 Z + Q
Yeh do exponent types hain jo beginners se darte hain, isliye proof mein inke use hone se pehle inhe visually anchor karte hain.
x − m x − m = x m 1 flip karne ka instruction hai, number ko negative banana nahi. x = 2 2 − 3 = 8 1
x p / q x p / q = ( q x ) p q p q kitne equal factors multiply ho ke x ; jaise x 1/2 = x x ⋅ x = x
Figure mein notice karo ki x 1/2 0 x 2 Steepness mein yeh difference exactly wahi hai jo derivative measure karta hai — yeh baat yaad rakho.
Definition Slope — ek line ki steepnessDo points ke beech, slope = run rise = change in input change in output 0
Intuition "Slope" poora game kyun hai"Jab main input ko nudge karta hun toh output kitni tezi se move karta hai?" yahi slope ka sawal hai. Derivative bas ek curve ka slope ek single point par hai — wahan curve ki apni direction ki steepness.
Lekin ek curve ki steepness point-to-point badlti rehti hai. Ise ek jagah pin karne ke liye hume limit chahiye.
h h x x + h h bahut chhota hona chahiye, lekin kabhi exactly 0
Figure mein, amber chord x x + h h flat na ho jaye — woh final direction tangent hai.
h → 0 lim lim h → 0 ( stuff ) h 0 0 stuff kis single number ki taraf approach karta hai? Hum h = 0 0 0 approach karke dodge karta hai, arrive karke nahi.
Intuition Algebra ki jagah limit kyun?Har fixed h chord ka slope deta hai (curve ke paas ek shortcut), jo thoda galat hota hai. Sirf limiting chord — ek single point par touch karne ke liye swing kiya gaya — true instantaneous slope hai. Proof h phir h → 0 Limit definition of the derivative mein hai.
( x + h ) n n ( x + h ) n finite sum milta hai:
( x + h ) n = x n + n x n − 1 h + ( 2 n ) x n − 2 h 2 + ⋯ + h n . x h
( k n ) ( k n ) n k n k h ( 1 n ) = n h
Intuition Positive integers ke liye specifically YEH tool kyun?Binomial theorem woh ek tool hai jo "h 0 x n h 1 n x n − 1 h h 2 n x n − 1 sirf tab jab expansion rokta hai , yaani sirf non-negative integer n n = 2 1 n = − 3 Binomial theorem .
Definition Fractions combine karnaA 1 − B 1 = A B B − A
Intuition Proof ko yeh kyun chahiyex − m = x m 1 fractions ka difference hai. Inhe common denominator ( x + h ) m x m x m − ( x + h ) m Quotient rule same result tak ek alternative route hai.
Definition Implicit differentiationAgar y x y = x dono sides ko term by term differentiate karte ho, y x y = x p / q q y q = x p dono sides integer powers hain !), phir differentiate karte hain.
Definition Chain rule (woh part jo tumhe bhoolna nahi chahiye)Jab tum kisi aisi cheez ko differentiate karte ho jo y y x d x d y d x d ( y q ) = q y q − 1 ⋅ d x d y . d x d y y
Common mistake Classic Stage-4 slipd x d y q = q y q − 1 bina d x d y lagta hai, lekin y x Chain rule aur Implicit differentiation .
Number types: positive int, zero, negative, rational
Binomial theorem and choose
Implicit diff plus chain rule
Khud test karo — jawab dene ke baad hi reveal karo.
Plain words mein x n Multiply x n x n
Convention se x 0 1
x − 3 x 3 1
x 2/3 x ka cube root, squared:
( 3 x ) 2 .
Kaun sa number-set symbol kehta hai "binomial theorem allowed hai"? Z +
f h f ( x + h ) − f ( x )
Geometrically, difference quotient kya hai? x x + h
Difference quotient mein hum h = 0 Yeh 0 0 0 approach karta hai.
Limit ke roop mein derivative ki definition? f ′ ( x ) = lim h → 0 h f ( x + h ) − f ( x )
( x + h ) n ( 1 n ) = n n x n − 1 h
n = 2 1 Expansion ek infinite series ban jaati hai, isliye tum divide-and-cancel cleanly nahi kar sakte.
A 1 − B 1 A B B − A
y q x y x q y q − 1 d x d y d x d y
