Question

Limits and Derivatives. Differentiate Sin x using the first principle.​

Answer 1

Hey!
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★Limits and Derivatives ★
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→ Differentiate Sin x using the first principle.

=> Given , f ( x ) = sin x

Then , f' ( x ) = Lim. h → 0 { f ( x + h ) - f ( x ) / h }

=> Lim.h→0 sin ( x + h ) - sin x / h

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★ we can use the identity

❗ Sin x - Sin y = 2 Cos( x + y /2) Sin( x - y/2 ) ❗

=> Lim h →0 2 Cos (x + x + h /2) Sin (x + h - x /2)

=> Lim.h→0 2 Cos ( 2x + h/2) Sin( h/2)

=> Lim.h→0 cos ( x + h/2 ) Lim. h→0 Sin (h/2)/ h/2

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★Now , Lim.h→0 ( Sin x /x ) => 1

•°• Lim.h → 0 Cos ( x + h/2 )

➡Put h = 0 , we Will get,

=> d / dx ( sin x ) = cos x ✔

➖•The derivative of sin x w.r.t x is => cos x ✔

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Answer 2

plz refer to the attachment for answer

here

thanks