Question

Find dy/ dx, if x 2/3 + y 2/3 = a 2/3 .

Answer 1

x^2/3+y^2/3=a^2/3
differentiate w.r.t x
2/3.x^(2/3-1)+2/3.y^(2/3-1)dy/dx=0 ( because a is constant )
now
2/3x^-1/3+2/3y^-1/3dy/dx=0
dy/dx=-(x/y)^-1/3

Answer 2

${x}^{ \frac{2}{3} } + {y}^{ \frac{2}{3} } = {a}^{ \frac{2}{3} } \\ differentiate \: \: w.r.t \: \: \: x \\ \frac{2}{3} {x}^{ - \frac{1}{3} } + \frac{2}{3} {y}^{ - \frac{1}{3} } \frac{dy}{dx} = 0 \\ \implies {y}^{ - \frac{1}{3} } \frac{dy}{dx} = - {x}^{ - \frac{1}{3} } \\ \implies \frac{dy}{dx} = - \frac{ {x}^{ - \frac{1}{3} } }{ {y}^{ - \frac{1}{3} } } \\ \implies \frac{dy}{dx} = - \frac{ {y}^{ \frac{1}{3} } }{ {x}^{ \frac{1}{3} } } \\ \implies \frac{dy}{dx} = - \sqrt[3]{ \frac{y}{x} }$