differentiation sinx
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y = sin x
dy / dx = Lim Δx -> 0 Δy/Δx
= Lim Δx -> 0, [ Sin (x + Δx) - Sin x ] / (x+Δx - x)
= 2 Cos ( x + Δx/2 ) Sin Δx/2 / Δx
= lim Δx -> 0, Cos ( x + Δx/2 ) * Sin (Δx/2) / (Δx/2)
as Δx -> 0, Δx/2 -> 0, then Sin (Δx/2) / (Δx/2) = 1
dy/dx = lim Δx/2 ->0 Cos (x + Δx/2) = Cos x
dy / dx = Lim Δx -> 0 Δy/Δx
= Lim Δx -> 0, [ Sin (x + Δx) - Sin x ] / (x+Δx - x)
= 2 Cos ( x + Δx/2 ) Sin Δx/2 / Δx
= lim Δx -> 0, Cos ( x + Δx/2 ) * Sin (Δx/2) / (Δx/2)
as Δx -> 0, Δx/2 -> 0, then Sin (Δx/2) / (Δx/2) = 1
dy/dx = lim Δx/2 ->0 Cos (x + Δx/2) = Cos x
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