Question

$\frac{1}{\sqrt{x+9}-\sqrt{x}}$ differentiate w.r.t $x$

Answer 1

Let $y=\frac{1}{\sqrt{x+9}-\sqrt{x}}$

$y=\frac{\sqrt{x+9}+\sqrt{x}}{(\sqrt{x+9}-\sqrt{x})(\sqrt{x+9}+\sqrt{x})}$

$y=\frac{\sqrt{x+9}+\sqrt{x}}{9}$

Differentiating both sides wrt x we get.

$\frac{dy}{dx}$ $=\frac{1}{9}\frac{1}{2\sqrt{x+9}}+\frac{1}{2\sqrt{x}}$

$\frac{dy}{dx}=\frac{\sqrt{x}+\sqrt{x+9}}{18\sqrt{x(x+9)}}$