Why do you blame the Product Rule?

(4x-1) * (4x-1)^0.5

First, the easy way.

(4x-1) * (4x-1)^(1/2) = (4x-1)^(3/2)

(d/dx)[(4x-1)^(3/2)] = [(3/2)*(4x-1)^(1/2)]*4 = 6*(4x-1)^(1/2)

Now the Product Rule

(d/dx)[(4x-1) * (4x-1)^(1/2)] = (4x-1)*(1/2)*[(4x-1)^(-1/2)]*4 + [(4x-1)^(1/2)]*4

= 2*(4x-1)^(1/2) + 4*(4x-1)^(1/2) = 6*(4x-1)^(1/2)

That looks very similar to the other version.