Verified answer

y = [x + (x^2 + 24)^(1/2)]^(1/3)

y = t^1/3

t = x + (x^2 + 24)^(1/2)

dy/dt = (1/3)t^(-2/3)

dt/dx =1 +(1/2)(x^2 + 24)^(-1/2)d/dx(x^2 + 24)

dt/dx =1 +(1/2)(x^2 + 24)^(-1/2)(2x)

dt/dx =1 +x(x^2 + 24)^(-1/2)

dy/dx =( dy/dt)(dt/dx)

dy/dx = (1/3)t^(-2/3) [1 +x(x^2 + 24)^(-1/2)]

dy/dx = (1/3)[ x + (x^2 + 24)^(1/2)]^(-2/3) [1 +x(x^2 + 24)^(-1/2)]