Question

Find $\frac{dy}{dx}$ , if $x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$

Answer 1

Consider the given equation.
$x^{\frac{2}{3}}+y^{\frac{2}{3}}=a^{\frac{2}{3}}$ ……. (1)

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

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

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

$\frac{1}{x^{\frac{1}{3}}}+\frac{1}{y^{\frac{1}{3}}}\frac{dy}{dx}=0$

$\frac{1}{y^{\frac{1}{3}}}\frac{dy}{dx}=-\frac{1}{x^{\frac{1}{3}}}$

$\frac{1}{y^{\frac{1}{3}}}\frac{dy}{dx}=-\frac{1}{x^{\frac{1}{3}}}$

$\frac{dy}{dx}=-\frac{y^{\frac{1}{3}}}{x^{\frac{1}{3}}}$

$\frac{dy}{dx}=-{\left(\frac{y}{x}\right)}^{\frac{1}{3}}$

Hence, this is the answer.