Explain the chain rule in differential calculus with some examples to make it clear :)
Answers
Dekho , I will use hindi to explain it more simply..
Chain rule hum tab use krte hain jab koi complex function ho. Complex function bole to function ke andar function .
Jaise ki (sinx)² , log|sinx| etc..
Abb tum sochoge ki pta kaise chalega ki complex function hai...
To simple funda hai...
Example 1 :
⇒ y = log|sinx| ka example lete hain...
⇒ agr logx hota to uska differentiation 1/x hota .. aur agr sirf sinx hota to uska differentiation sirf cosx hota . but yaha pe log ke andar sinx hai . isliye ye ek complex function hai .
Abb isko jaldi dimaag mei solve krne ke liye ye cheez krna : pehle log ka differentiation phir sin ka differentiation . kyunki logx mei x ki jagah sinx rkha hua hai to differentiation mei bhi x ki jagah sinx hoga .
dy/dx = 1/sinx × cosx = cosx/sinx = cotx .
Isi tarah se (sinx)³ solve hoga . ismein power ke andar ek function hai .
to isko hum aise likh skte hain ;
= 3(sinx)³⁻¹ × d/dx(sinx) = 2sin²x (cosx) ... {power rule to aata hoga}
Note : Agr koi trigonometric function hai jaise ki sin(cosx) . iska differentiation abb kaise krenge ...
dekho , ismein bhi kuch wahii funda hai...pehle sin ka differentiation krro ...
prr dhyan se dekho kyunki yaha pe x ki jagah cosx hai to sin ka differentiation cos but x ki jagah yaha pe bhi cosx aayega...
mtlb sin(cosx) = cos(cosx) × d/dx(cosx) ...
abhi ye aadha complete ho gya...ismein aage cos ka differentiation krenge...
sin(cosx) = cos(cosx) × (-sinx) = -sinx cos(cosx)
.. yaha pe cos(cosx) mei cos ke andar cosx uska angle hai na ki function ...
abb ek chota sa mantra yaad kro aur shuru ho jaao ques krna ...
MANTRA : bahar ka differentiation {x ki jagah pe andar wala likhna hai} × andar ka differentiation .