Question

$y=\sqrt{\frac{x}{a}+}\sqrt{\frac{a}{x},}provethat2xy\frac{dy}{dx}=\left(\frac{x}{a}-\frac{a}{x}\right)$

Answer 1

We have,

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

Squaring both sides,

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

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

On differentiating w.r.t $x$, we get

$2y\frac{dy}{dx}=\frac{1}{a}-\frac{a}{x^2}+0$

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

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

Hence, proved.