Differentiate each of the following from first principle: `sqrt(sin2x)`

Explanation

`f'(x)=lim_(h->0)frac{f(x+h)-f(x)}{h}`

`=lim_(h->0)frac{sqrt(sin(2x+2h))-sqrt(sin 2x)}{h} xx frac{sqrt(sin(2x+2h))+sqrt(sin 2x)}{sqrt(sin(2x+2h))+sqrt(sin 2x)}`

`=lim_(h->0)frac{sqrt(sin(2x+2h))-sqrt(sin 2x)}{h(sqrt(sin(2x+2h))+sqrt(sin 2x))}`

`=lim_(h->0) frac{2 cos(frac{2x+2h+2x}{2})sin(frac{2x+2h-2x}{2})}{h(sqrt(sin(2x+2h))+sqrt(sin 2x))}`

`=lim_(h->0)frac{2 cos(2x+h) sin h}{h (sqrt(sin(2x+2h))-sqrt(sin 2x))}`

`=lim_(h->0)2 cos(2x+h) lim_(h->0) frac{sin h}{h} lim_(h->0) frac{1}{sqrt(sin(2x+2h))+sqrt(sin 2x)}`

`=frac{2 cos 2x}{2 sqrt(sin 2x)}=>frac{cos 2x}{sqrt(sin 2x)}`