Given that y = 3x(^2) + 6x(^1/3) + (2x(^3) - 7)/(3(sqrt(x))) when x > 0 find dy/dx

Firstly, the (2x(^3) - 7)/(3(sqrt(x))) can be split into (2x(^3))/(3(sqrt(x)) and -7/(3(sqrt(x)). These can then be simplified to (2/3)x(^5/2) and -(7/3)x(^-1/2) respectively. This then gives the equation y = 3x(^2) + 6x(^1/3) + (2/3)x(^5/2) - (7/3)x(^-1/2).

By multiplying the coefficients of x by the power of x and then taking 1 from the power it is found that dy/dx = 6x + 2x(^-2/3) + (5/3)x(^3/2) + (7/6)x(^-3/2).