Find `(dy)/(dx)`, if `x^(2/3)+y^(2/3)=a^(2/3)`.

Explanation

Let` x = a cos^(3) theta`
` y = a sin^(3) theta `
then ,
` x^(2//3) + y^(2//3) = (a cos^(3) theta)^(2//3) + (a sin^(3) theta)^(2//3)`
` = a^(2//3) [ cos^(2) theta + sin^(2) theta] = a^(2//3)`
Hence, ` x = a cos^(3) , y = a sin^(3) theta` is parametric equation of ` x^(2//3) + y^(2//3) = a^(2//3)`
Now, ` (dx)/(dtheta) = -3 a cos^(2) theta sin theta`
` and (dy)/(dtheta) = 3 a sin^(2) theta cos theta `
`(dy)/(dx) = (dy//d theta)/(dx // d theta) = (3 a sin^(2) theta cos theta)/(-3a cos^(2) theta sin theta)`
` - tan theta - root(-3)(y/x)`