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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