Find the derivative of `sqrt(tanx)` using first principles.

Explanation

`f(x) = sqrt(tanx)`

`Lim_(h->0) (f(x+h)-f(x))/h = Lim_(h->0) (sqrt(tan(x+h))-sqrt(tanx))/h`

Multiplying nunerator and denominator by `(sqrt(tan(x+h))+sqrt(tanx))`,

` =Lim_(h->0) (tan(x+h) - (tanx))/ ((sqrt(tan(x+h))+sqrt(tanx))h)`

`=Lim_(h->0) ((sin h)/(cos(x+h)cosx))/ ((sqrt(tan(x+h))+sqrt(tanx))h)`

`= Lim_(h->0) sin h/h xx Lim_(h->0)(1/((cos(x+h)cosx)(sqrt(tan(x+h))+sqrt(tanx))))`

`=1 xx 1/(2cos^2xsqrt(tanx))`

`=sec^2x/(2sqrttan x)`