Differentiate each of the following from first principle: `sqrt(2x^2+)1`

Explanation

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

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

`=lim_(h->0)frac{sqrt(2x^2+2h^2+4h+1)-sqrt(2x^2+1)}{h}xx frac{sqrt(2x^2+2h^2+4h+1)+sqrt(2x^2+1)}{sqrt(2x^2+2h^2+4h+1)+sqrt(2x^2+1)}`

`=lim_(h->0) frac{2h+4x}{sqrt(2x^2+2h^2+4h+1)-sqrt(2x^2+1)}`

`=frac{4x}{2 sqrt(2x^2+1)}=>frac{2x}{ sqrt(2x^2+1)`