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

Explanation

माना `y=sin^(2)x .........(i)`
`therefore y+deltay=sin^(2)(x+deltax)` .......(ii)
`therefore y+deltay-y=sin^(2)(x+deltax)-sin^2x`
`Rightarrow deltay=sin(x+deltax+x)sin(x+deltax-x)`
`deltay=sin(2+deltax).sindeltax`
दोनों पक्षों को `deltax` से भाग देकर `underset(deltax to 0)` लेने पर
`underset(deltax to 0)lim (deltay)/(deltax)=(dy)/(dx)=underset(deltax to 0)lim (sin (2x+deltax)sin deltax)/(deltax)`
`=underset(deltax to 0)lim sin(2x+deltax).underset(deltax to 0)lim (sindeltax)/(deltax) `
`=sin(2x+0) xx 1" "(therefore underset(theta to 0)lim (sin theta)/(theta)=1)`
`=sin 2x`
`therefore (d)/(dx)(sin^(2)x)=2sin x cos x`