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Integration of sin x from first principles

P
Pramod .
18/10/2021 0 0

y' = Lt   [f(x+∆x) - f(x)]/∆x

      ∆x => 0

Therefore if y = sin x

y' = Lt  [sin (x + ∆x) - sin x] / ∆x

       ∆x=>0    

y' = Lt [sin x cos ∆x+cos x sin ∆x - sin x]/∆x

     ∆x => 0

=>Lt [ sin x cos ∆x - sin x + cos x sin ∆x]/∆x

   ∆x => 0

=> Lt [ sin x (cos ∆x - 1) + cos x sin ∆x]/ ∆x

    ∆x => 0s

=>sin x Lt(cos ∆x-1)/∆x+cos xLt(sin ∆x/∆x)

              ∆x => 0                         ∆x=> 0

=> sin x • 0 + cos x• 1

=> dy/dx (sin x) = 0 + cos x

=> dy/dx (sin x ) = cos x

 

 

 

[ Lt sin x/x = 1 ] & Lt (cos x - 1)/x = 1-1 = 0]

    x => 0                      x => 0

 

 

 

 

 

 

 

 

 

 

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