प्रथम सिद्धांत से `cos^(2) x` का अवकल गुणांक ज्ञात कीजिएः

Explanation

माना `y=cos^(2)x`
`Rightarrow y+deltay=cos^(2)(x+deltax)`
`Rightarrow y+deltay-y=cos^(2)(x+deltax)-cos^(2)x`
`Rightarrow (deltay)/(deltax)=(cos^(2)(x+deltax)-cos^(2)x)/(deltax)`
अब `(dy)/(dx)=underset(deltax to 0)lim (deltay)/(deltax)`
`=underset(deltax to 0)lim (cos^(2)(x+deltax)-cost^(2)x)/(deltax)`
`=underset(deltax to 0)lim ([cos^(2)(x+deltax)+cosx].[cos(x+deltax)-cosx])/(deltax)`
`=underset(deltax to 0)lim ([2cos (x+(deltax)/(2))cos((deltax)/(2))][-2sin (x+(deltax)/(2))sin((deltax)/(2))])/(deltax)`
`=underset(deltax to 0)lim cos(x+(deltax)/(2)).underset(deltax to0)lim cos ((deltax)/(2)).underset(deltax to0)lim sin(x+(deltax)/(2)).underset(deltax to0)lim (sin((deltax)/(2)))/(((8delta)/(2)))`
`=-2cos x xx 1 xx sin xx 1`
`=sin 2x`