Question

find the derivation of y with respect to x

x^2/3 +y^2/3 = pi^2/3

2/3x^(-1/3)+2/3y^(-1/3) dy/dx = 0 how do I solve this?

I also need help with e^(xy) = cos(y^4)

Answer 1

2/3x^(-1/3) + 2/3y^(-1/3) dy/dx
+2/3y^(-1/3) dy/dx = -2/3x^(-1/3)
dy/dx = -(x/y)^(-1/3) = -∛(y/x)

Just keep in mind the chain rule and the product rule
e^(xy) = cos(y^4)
e^(xy) (y + xy') = -sin(y^4) * 4y^3 y'
form here on, it's just algebra...
e^(xy) * xy' + 4y^3 sin(y^4) y' = -y e^(xy)

y' = -(y e^(xy)) / (x e^(xy) + 4y^3 sin(y^4))