Question

If $y=\sqrt{\frac{x}{a}}+\sqrt{\frac{a}{x}}$, prove that $2xy\frac{dy}{dx}=\frac{x}{a}-\frac{a}{x}$

Answer 1

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

Squaring both sides

$y^2=\frac{x}{a}+\frac{a}{x}+2$

$2y\frac{dy}{dx}=\frac{1}{a}-\frac{a}{x^2}$

Multiplying by $x$

$2xy\frac{dy}{dx}=\frac{x}{a}-\frac{a}{x}$