# How do you differentiate x^(2/3)+y^(2/3)=pi^(2/3)?

May 4, 2018

Differential is $- \sqrt[3]{\frac{x}{y}}$

#### Explanation:

Differential of ${x}^{n}$ is $n {x}^{n - 1}$ and that of ${y}^{n}$, is $n {y}^{n - 1} \frac{\mathrm{dy}}{\mathrm{dx}}$ using implicit differentiation. Also as $\pi$ is a constant, its differential, or that of any of its power, is $0$

Hence differential of ${x}^{\frac{2}{3}} + {y}^{\frac{2}{3}} = {\pi}^{\frac{2}{3}}$

is $\frac{2}{3} {x}^{- \frac{1}{3}} + \frac{2}{3} {y}^{- \frac{1}{3}} \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

and $\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{\frac{2}{3} {y}^{- \frac{1}{3}}}{\frac{2}{3} {x}^{- \frac{1}{3}}} = - \frac{\frac{1}{\sqrt[3]{y}}}{\frac{1}{\sqrt[3]{x}}}$

= $- \sqrt[3]{\frac{x}{y}}$