Final answer:
To find the derivative of the function (6x + 1)^(1/2) / (3x - 2)^4, the quotient rule is applied, derivatives of the numerator and the denominator are found, and subtraction and simplification are performed. The correct answer is option B.
Step-by-step explanation:
The derivative of the given function (6x + 1)^(1/2) / (3x - 2)^4 can be found using the quotient rule in calculus which is given by:
(v'(x)u(x)-u'(x)v(x)) / v(x)^2 where u(x) is the numerator and v(x) is the denominator.
Let u = (6x + 1)^(1/2) and v = (3x - 2)^4. First, we find the derivative of u, u'. The derivative of (6x + 1)^(1/2) is (1/2)(6x + 1)^(-1/2)*6. Next, find the derivative of v, v', which is 4(3x - 2)^3*3.
Apply the quotient rule:
u' = (1/2)(6x + 1)^(-1/2)*6
v' = 4(3x - 2)^3*3
u'v - uv' = (3(6x + 1)^(-1/2))(3x - 2)^4 - (6x + 1)^(1/2)*12(3x - 2)^3
Now simplify and factor out common terms and then divide by v^2, which is (3x - 2)^8.
The simplified derivative leads us to either A or B as the correct option, and the correct sign can be determined by carefully tracking the sign throughout the calculation process. The correct answer is B, as it maintains the negative sign from the term -uv'.