Can anyone derive this correctly with the correlation of y's derivative being (dy/dx):

derive(y^3):

Let's say y = x^n. Now, the derivative of y (y' or dy/dx, however you like to write it) would be n * x ^(n-1). To derive y^3, you would bring the 3 out to the front and subtract the exponent 3 (cube) by one. So, dy/dx = 3*y^2.