Differentiate each of the following from the first principle.
(i) `sqrt(sinx)` , (ii) `sqrt(cos x)`
(iii) `sqrt(tanx)` , (iv) `sqrt(cosecx)`

Explanation

(i) Let y = `sqrt(sinx)`.
Let `deltay` be an increment in y, corresponding to an increment `deltax` in x.
Then, `y + deltay = sqrt(sin(x+deltax))`
`rArr deltay = sqrt(sin(x+deltax))-sqrt(sinx)`
`rArr (deltay)/(deltax) =(sqrt(sin(x+deltax))-sqrt(sinx))/(deltax)`
`rArr (dy)/(dx) = underset(deltararr 0)("lim") (deltay)/(deltax)`
`= underset(deltax rarr 0)("lim")(sqrt(sin(x+delta))-sqrt(sinx))/(deltax)`
`= underset(deltax rarr 0)("lim")({sinsqrt(x+deltax)-sqrt(sinx)})/(deltax)`
`= underset(deltax rarr 0)("lim")({sin(x+deltax)-sinx})/(deltax.{sqrt(sin(x+deltax)) + sqrt(sinx}})`
`= underset(deltaxrarr0)("lim")(2cos(x+(delta)/(2))sin((delta)/(2)))/({sqrt(sin(x+deltax)) +sqrtsinx}deltax)`.
`=underset(deltax rarr 0)("lim")cos(x+(deltax)/(2)). underset(deltax rarr 0)("lim") (sin((deltax)/2))/(((deltax)/(2)))`
`underset(deltax rarr 0)("lim")1/({sqrt(sin(x+deltax)) + sqrt(sinx)])`
`= {cosx.1.1/(2sqrt(sinx))} = (cosx)/(2sqrt(sinx))`.
Hence, `d/(dx) (sqrt(sinx)) = (cos x)/(2sqrt(sinx))`.
(ii) Let `y = sqrt(cos x)`.
Let `deltay` be an increment in y, correponding to an increment `deltax` in x.
Then, `y + deltay = sqrt(cos(c+deltax))`
`rArr deltay = sqrt(cos(x+deltax))-sqrt(cosx)`
`rArr (deltay)/(deltax) = ({sqrt(cos(x+deltax))-sqrt(cosx)})/(deltax)`
`rArr (dy)/(dx) =underset(deltax rarr 0)("lim") ({sqrt(cos(x+deltax))-sqrt(cosx)})/(deltax)`
`= underset(deltaxrarr0)("lim")({sqrt(cos(x+deltax))-sqrt(cosx)})/(deltax) xx ({sqrt(cos(x+deltax)) + sqrt(cosx)})/({sqrt(cos(x+deltax)) + sqrt(cosx)})`
`=underset(deltaxrarr0)("lim")({cos(x+deltax) - cos x})/(deltax.{sqrt(cos(x+deltax))+ sqrt(cosx)})`
`= underset(deltax rarr 0)("lim")(2sin(x+(deltax)/(2))sin((deltax)/(2)))/(deltax.{sqrt(cos(x+deltax)) + sqrt(cosx)})`
`[ :' cos C - cos D = - 2sin ((C+D)/(2)) sin((C-D)/(2))]`
`=underset(deltax rarr 0)(-"lim")sin(x+(deltax)/(2)).underset(deltax rarr 0)("lim")(sin(deltax//2))/((deltax//2))`.
`underset(deltax rarr 0)("lim")(1)/({sqrt(cos(x+deltax))+sqrt(cosx)})`
`= (-sinx) xx 1 xx 1/(2sqrt(cosx))`
Hence, `(d)/(dx) (sqrt(cosx)) = (-sinx)/(2sqrt(cosx))`
(iii) Let `y = sqrt(tanx)`.
Let `deltay` be an increment in y. corresponding to an increment `deltax` in x.
Then, `y + deltax = sqrt(tan(x + deltax))`
`rArr deltay = sqrt(tan(x+deltax)) - sqrt(tanx)`
`rArr (deltay)/(deltax) = (sqrt(tan(x+deltax)) - sqrt(tanx))/(deltax)`
`rArr (dy)/(dx) = underset(deltax rarr 0)("lim") (deltay)/(deltax)`
`= underset(deltaxrarr0)("lim"){(sqrt(tan(x+deltax))- sqrt(tanx))/(deltax) xx (sqrt(tan(x+deltax))+sqrt(tanx))/(sqrt(tan(x+deltax))+sqrt(tanx))}`
`= underset(deltaxrarr0)("lim")(tan(x+deltax)-tanx)/(deltax[sqrt(tan(x+deltax))+sqrt(tanx)])`
`= underset(deltararr0)("lim")({(sin(x+deltax))/(cos(x+deltax))-(sinx)/(cosx)})/(deltax[sqrt(tan(x+deltax))+sqrt(tanx)])`
`= underset(deltaxrarr0)("lim")(sin(x+deltax)cosx - cos (x+deltax) sinx)/(cos(x+deltax) cosx. deltax. [sqrt(tan(x+deltax)) + sqrt(tanx)])`
`= 1/(cosx). underset(deltax rarr 0)("lim") (1)/(cos(x+deltax)) . underset(deltaxrarr0)("lim") (sindeltax)/(deltax).underset(deltararr0)("lim") (1)/((sqrt(tan(+deltax)) + sqrt(tanx)))`
`= ((1)/(cosx) . (1)/(cosx)1.(1)/(2sqrt(tanx)))= (sec^(2)x)/(2sqrt(tanx))`
Hence `d/(dx) (sqrt(tanx)) = (sec^(2)x)/(2sqrt(tanx))`.
Let `y = sqrt(cosecx)`.
Let `deltay` be an increment in y , correspondingto an increment `deltax` in x.
Then, `y +deltay = sqrt(cosec(x+deltax))`
`rArr deltay = sqrt(cosec(x+deltax)) - sqrt(cosecx)`
`rArr (dy)/(deltax) = (sqrt(cosec(x+deltax)) - sqrt(cosecx))/(deltax)`
`rArr (dy/(dx)= underset(deltax rarr 0)"lim" (dy)/(dx)`
`= underset(deltax rarr 0)("lim") (sqrt(cosec(x+deltax)) - sqrt(cosecx))/(deltax)`
`= underset(deltararr0)("lim")({(1)/(sqrt(sin(x+deltax))) -(1)/(sqrt(sinx))})/(deltax)`
`= underset(deltaxrarr0)("lim")({sqrt(sinx)-sqrt(sin(x+deltax))})/(deltax.sqrt(sin(x+deltax)).sqrt(sinx)) xx ({sqrt(sinx) + sqrt(sin(x+deltax))})/({sqrt(sinx) + sqrt(sin(x+deltax))})`
`= underset(deltararr0)("lim")({sinx-sin(x+deltax)})/({sqrt(sin(x+deltax)).sqrt(sin(x))}) xx (1)/(deltax.{sqrt(sinx) + sqrt(sin(x+deltax))})`
`= underset(deltararr0)("lim")(-2cosx(x+(deltax)/(2))sin((deltax)/(2)))/({sqrt(sin
(x+deltax))}.sqrt(sinx).deltax.{sqrt(sinx)+sqrt(sin(x+deltax))})`
`= - underset(deltax)("lim")cos (x+(deltax)/(2)).underset(deltaxrarr0)("lim") (sin ((deltax)/(2)))/(((deltax)/(2)))`
`underset(deltararr0)("lim") (1)/(sqrt(sin(x+deltax)).sqrt(sinx)).underset(deltararr0)("lim")(1)/({sqrt(sinx)+sqrt(sin(x+deltax))})`
`= - cos x xx 1 xx (1)/(sqrt(sinx). sqrt(sinx)).(1)/((sqrt(sinx) +sqrt(sinx)))`
`= (-cosx)/(sinx).(1)/(2sqrt(sinx)) = - 1/2 sqrt(cosecx) cot x`.
Hence, `d/(dx) (sqrt(cosecx)) = - 1/2 sqrt(cosecx) cotx`.