Question

If y=√x/a+√a/x then prove that 2xy dy/dx
=x/a-a/x

Answer 1

$$\begin{gathered} \mathrm{y}=\sqrt{\frac{\mathrm{x}}{\mathrm{a}}}+\sqrt{\frac{\mathrm{a}}{\mathrm{x}}} \\ \mathrm{On}\mathrm{squaring}\mathrm{both}\mathrm{sides} \\ {\mathrm{y}}^2=\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{a}}{\mathrm{x}}+2\times \frac{\mathrm{x}}{\mathrm{a}}\times \frac{\mathrm{a}}{\mathrm{x}} \\ {\mathrm{y}}^2=\frac{\mathrm{x}}{\mathrm{a}}+\frac{\mathrm{a}}{\mathrm{x}}+2 \\ \mathrm{On}\mathrm{taking}\mathrm{differentiation}\mathrm{both}\mathrm{sides}, \\ 2\mathrm{y}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{1}{\mathrm{a}}-\frac{\mathrm{a}}{{\mathrm{x}}^2} \\ \mathrm{On}\mathrm{multiplying}\mathrm{by}\mathrm{x}\mathrm{both}\mathrm{sides} \\ 2\mathrm{xy}\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{\mathrm{x}}{\mathrm{a}}-\frac{\mathrm{a}}{\mathrm{x}} \end{gathered}$$