Question

Differentiate

$\frac{x+\sqrt{x^2+1}}{\sqrt{x^2+1}}$

Answer 1

Let

$y=\frac{x+\sqrt{x^2+1}}{\sqrt{x^2+1}}$

$y=\frac{x}{\sqrt{x^2+1}}+1$

Differentiate on both sides w.r.t x

$\frac{dy}{dx}=\frac{1\sqrt{x^2+1}-x\frac{2x}{2\sqrt{x^2+1}}}{(\sqrt{x^2+1}{)}^2}+0$

$=\frac{\sqrt{x^2+1}-\frac{x^2}{\sqrt{x^2+1}}}{(x^2+1)}$

$\therefore \frac{dy}{dx}=\frac{x^2+1-x^2}{(x^2+1{)}^{\frac{3}{2}}}=\frac{1}{(x^2+1{)}^{\frac{3}{2}}}$