y= (4x^2+7)^2 * (3x^3+1)^4............we need the chain rule and product rule here

=> y' = 2(4x^2 + 7)(8x)(3x^3 + 1)^4 + (4x^2+7)^2 * 4(3x^3 + 1)^3 * 9x^2

=> y' = 16x(4x^2 + 7)(3x^3 + 1)^4 + (36x^2)(4x^2+7)^2 * (3x^3 + 1)^3

=> y' = [4x(4x^2 + 7)(3x^3 + 1)^3]* [4(3x^3 + 1) + 9x(4x^2+7)]

y= (2x)^(1/2) + (2/x)^(1/2).......we need the chain rule for each of these

=> y' = (1/2)(2x)^(-1/2) * 2 + (1/2)(2/x)^(-1/2) * (-2x^-2)

=> y' = (2x)^(-1/2) - (x^-2)(2/x)^(-1/2)

.......product rule combined with chain rule again

y= (x^2-5)^(1/2) * (x^2+3)^(1/3)

=> y' = (1/2)(x^2 - 5)^(-1/2) * (2x)(x^2+3)^(1/3) + (x^2-5)^(1/2) * (1/3)(x^2 + 3)^(-2/3) * (2x)

=> y' = x(x^2 - 5)^(-1/2) * (x^2+3)^(1/3) + (2/3)x(x^2-5)^(1/2) * (x^2 + 3)^(-2/3)