Differentiate each of the following from first principle: `sinsqrt(2x)`

Explanation

`f(x) =sqrt{sin 2 x} `


because `quad f^{prime}(x) =lim _{h rightarrow 0} frac{f(x+h)-f(x)}{h} `


`=lim _{h rightarrow 0} frac{sqrt{sin 2(x+h)}-sqrt{sin 2 x}}{h}`



Now multipling `t` dividing by `(sqrt{sin 2(x+h)}+sqrt{sin 2 x})`



`=lim _{h rightarrow 0} frac{sqrt{sin 2(x+h)}-sqrt{sin 2 x}}{h} times frac{sqrt{sin 2(x+h)}+sqrt{sin 2 x}}{sqrt{sin 2(x+h)}+sqrt{sin 2 x}} `


`= lim _{h rightarrow 0} frac{sin (2 x+2 h)-sin 2 x}{h(sqrt{sin 2(x+h)}+sqrt{sin 2 x})} `


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