Differentiate each of the following from first principle: `sqrt(tanx)`

Explanation

`f(x)=sqrt(tanx)`


`f'(x)=lim_(h->0)(f(x+h)-f(x))/(h)`


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


`f'(x)=lim_(h->0)(sin(x+h)cos(x+h)-sinxcosx)/(hcos(x+h)cosx(sqrt(tan(x+h))+sqrt(tanx)))`


`f'(x)=lim_(h->0)(sin(x+h-x))/(hcos(x+h)cosx(sqrt(tan(x+h))+sqrt(tanx)))`


`f'(x)=lim_(h->0)(sin(h))/(hcos(x+h)cosx(sqrt(tan(x+h))+sqrt(tanx)))`


`f'(x)=lim_(h->0)(1)/(cos(x+h)cosx(sqrt(tan(x+h))+sqrt(tanx)))`


`f'(x)=lim_(h->0)(1)/(cos(x+0)cosx(sqrt(tan(x+0))+sqrt(tanx)))`


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