मूल नियमो से `sqrt(sin x)` का अवलंकन गुणांक ज्ञात कीजिएः

Options

A. `(cos x)/(5sqrtsin x)`
B. `(cos x)/(4sqrtsin x)`
C. `(cos x)/(3sqrtsin x)`
D. `(cos x)/(2sqrtsin x)`

Answer

D

Explanation

माना `y=sqrt(sin x)` .......(i)
`therefore y+deltay=sqrt(sin(x+deltax))` .......(ii)
`therefore y+deltay-y=sqrt(sin (x+deltax))-sqrt(sin x)`
`Rightarrow deltay=sqrt(sin (x+deltax))-sqrt(sin x)`
अंश व हर को `[sqrt(sin (x+deltax))+sqrt(sin x)]` से गुना करने पर,
`deltay=(sin (x+deltax)-sin x)/(sqrt(sin (x+deltax))+sqrt(sin x))`
`=deltay=(2cos (x+(deltax)/(2))sin((deltax)/(2)))/(sqrt(sin x(x+deltax))+sqrt(sin x))`
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 (2 cos (x+(deltax)/(2)))/(sqrt(sin (x+deltax))+sqrt(sin x)).(sin""((deltax)/(2)))/(2(deltax)/(2))`
`=underset(deltax to 0)lim (cos (x+(deltax)/(2)))/(sqrt(sin (x+deltax))+sqrt(sin x)).underset(deltax to 0)""(sin""((deltax)/(2)))/(((deltax)/(2)))`
`=(cos (x+0))/(sqrt(sin (x+0)+sqrt(sin x)))xx1 (therefore underset(theta to0)lim (sin theta)/(2)=1)`
`=(cos x)/(2sqrt(sin x))`
`=therefore (d)/(dx)sqrt((sin x))=(cos x)/(2sqrtsin x)`