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If y=sin(sqrt(sinx+cosx)), find (dy)/(dx...

If `y=sin(sqrt(sinx+cosx)),` find `(dy)/(dx)`.

A

`cos (sqrt(sinx + cosx))`

B

`(cos (sqrt(sinx + cosx)) (cosx - sinx))/(2sqrt(sinx - cosx))`

C

`(cos (sqrt(sinx + cosx)) (cosx - sinx))/(2sqrt(sinx + cosx))`

D

`(cos (sqrt(sinx + cosx)) (cosx - sinx))`

Text Solution

Verified by Experts

The correct Answer is:
C
Putting `(sinx + cosx) =t` and `sqrt((sinx + cosx)) = sqrt(t) = u`, we get
`y = sin u, u = sqrt(t)` and `t = (sinx + cosx)`.
`:. (dy)/(du) = cos u, (du)/(dt) = 1/2 t^(-1//2) = (1)/(2sqrt(t))`
and `(dt)/(dx) = (cos x - sinx)`.
So, `((dy)/(du) xx (du)/(dt) xx (dt)/(dx)) = (cos u)/(2sqrt(t)). (cos x - sinx)`
`= (cossqrt(t))/(2sqrt(t)).(cosx - sinx) [:' u = sqrt(t)]`
`= (cos (sqrt(sinx + cosx)) (cosx - sinx))/(2sqrt(sinx + cosx)) [ :' t = (sinx + cosx)]`.
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