4.1.16
Intuition Core idea (WHY yeh exist karta hai)Ek composite function f ( g ( x )) machine ke andar machine hai. Tum x g g ′ ( x ) f us wiggle ko apni rate f ′ ( g ( x )) multiply hote hain kyunki yeh sequence mein act karte hain, jaise ek gearbox mein gear ratios hote hain.
Agar g x f g ( x ) h ( x ) = f ( g ( x )) x h ′ ( x ) = f ′ ( g ( x )) ⋅ g ′ ( x ) . u = g ( x ) y = f ( u ) d x d y = d u d y ⋅ d x d u .
Leibniz form mein multiplication cancelling fractions jaisi lagti hai — yeh ek mnemonic hai, proof nahi (yeh literal fractions nahi hain).
Common mistake Steel-man: "Δ u Kyun sahi lagta hai: naïve cancellation se correct answer milta hai, aur d y / d x = ( d y / d u ) ( d u / d x ) Kyun galat hai: Δ u Δ x = 0 0 g ( x ) = x 2 sin ( 1/ x ) 0 Δ u Fix: E ( k ) Δ u = 0
h ( x ) = ( 3 x 2 + 1 ) 5 f ( u ) = u 5 g ( x ) = 3 x 2 + 1 f ′ ( u ) = 5 u 4 kyun? outer shell par power rule.
g ′ ( x ) = 6 x kyun? inside ki derivative.
h ′ ( x ) = 5 ( 3 x 2 + 1 ) 4 ⋅ 6 x = 30 x ( 3 x 2 + 1 ) 4 . Yeh step kyun? u = g ( x ) f ′ ( u ) g ′ ( x )
h ( x ) = sin ( x 2 ) f ( u ) = sin u ⇒ f ′ ( u ) = cos u g ( x ) = x 2 ⇒ g ′ ( x ) = 2 x h ′ ( x ) = cos ( x 2 ) ⋅ 2 x = 2 x cos ( x 2 ) . Yeh step kyun? Outer sin phir inside ki rate se multiply karo. Note karo cos ( x 2 ) = cos ( 2 x )
h ( x ) = e s i n x f ( u ) = e u , f ′ ( u ) = e u g ( x ) = sin x , g ′ ( x ) = cos x h ′ ( x ) = e s i n x cos x .
Worked example 4. Triple nesting:h ( x ) = cos ( 4 x ) = ( cos 4 x ) 1/2 cos 4 x h ′ ( x ) = 2 1 ( cos 4 x ) − 1/2 ⋅ ( − sin 4 x ) ⋅ 4 = c o s 4 x − 2 s i n 4 x . Yeh step kyun? Har layer ek factor contribute karti hai; saare layer-rates ko multiply karo (onion ko outside-in peel karo).
Worked example 5. Ek known rule ke against verify karo —h ( x ) = ( x 2 ) 3 = x 6 6 ( x 2 ) 2 ⋅ 2 x = 6 x 4 ⋅ 2 x = 12 x 5 d x d x 6 = 6 x 5 Mismatch?! Re-check karo: outer f ( u ) = u 3 f ′ = 3 u 2 3 ( x 2 ) 2 ⋅ 2 x = 3 x 4 ⋅ 2 x = 6 x 5 Yeh step kyun? Forecast-then-verify ne meri arithmetic slip pakdi (maine f ′ 3 6 f ′
Recall Feynman: ek 12-saal ke bacche ko samjhaoSocho ek bike jisme do gears connected hain. Tum thoda pedal karte ho (yeh hai x 5 × 3 = 15 total speed-up paane ke liye, har gear ki speed-up ko multiply karo. f ′ ( g ( x )) g ′ ( x )
Mnemonic "Outside-in, peel the onion"D erivative of the O utside (inside ko akela chhodo), T imes derivative of the I nside.
DOTI : "Do the Outside, Then the Inside (aur multiply karo)."
Recall Active recall checkpointAnswers cover karo. Kya tum (a) dono notations state kar sakte ho, (b) explain kar sakte ho kyun naïve cancellation fail hoti hai, (c) bina dekhke sin ( x 2 )
What is the chain rule for h = f ( g ( x )) h ′ ( x ) = f ′ ( g ( x )) g ′ ( x )
Chain rule ko Leibniz notation mein state karo. d x d y = d u d y ⋅ d x d u u = g ( x )
Proof mein hum simply Δ u Δ u = g ( x + Δ x ) − g ( x ) 0 Δ x = 0
g Δ x → 0 Δ u → 0 g g
Rigorous proof mein E ( k ) Error E ( k ) = k f ( u + k ) − f ( u ) − f ′ ( u ) E ( 0 ) = 0 multiply karne deta hai, Δ u = 0
sin ( x 2 ) 2 x cos ( x 2 )
e s i n x e s i n x cos x
( 3 x 2 + 1 ) 5 30 x ( 3 x 2 + 1 ) 4
Triple nesting mein kitne factors aate hain? Ek per layer; saari layer-derivatives ko multiply karo, har outer ko relevant inner value par evaluate karke.
sin ( x 2 ) cos ( 2 x ) cos ( x 2 ) 2 x
Derivative — limit definition — chain rule directly difference-quotient limit se build hoti hai.Continuity implies differentiability fails converse — hum g Product rule aur Quotient rule — sibling differentiation rules; aksar chain rule ke saath combine hote hain.Implicit differentiation — literally chain rule hai y ( x ) Related rates — physical applications jahan d t d y = d x d y d t d x Inverse function derivative — f ( f − 1 ( x )) = x
wiggle amplified in sequence
multiply and divide by delta u
h' = f' of g x times g' x
dy/dx = dy/du times du/dx
Fraction cancelling mnemonic
Derivative definition limit
