Question:
Find $\frac{d y}{d x}$ in each of the following:
$x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$
Solution:
We are given with an equation $x^{2 / 3}+y^{2 / 3}=a^{2 / 3}$, we have to find $\frac{d y}{d x}$ of it, so by differentiating the equation on both sides with respect to $x$, we get,
$\frac{2}{3} \frac{1}{\mathrm{x}^{1 / 3}}+\frac{2}{3} \frac{1}{\mathrm{y}^{1 / 3}} \frac{\mathrm{dy}}{\mathrm{dx}}=0$
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\mathrm{y}^{1 / 3}}{\mathrm{x}^{1 / 3}}$
Or we can write it as,
$\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{-\sqrt{\mathrm{a}^{2 / 3}-\mathrm{x}^{2 / 3}}}{\mathrm{x}^{1 / 3}}$