Final answer:
To find the derivative of the function f(x) = 3x^(3/6)/x^(2/3) using the quotient rule, differentiate the numerator and denominator separately and then apply the quotient rule formula.
Step-by-step explanation:
To find the derivative of the function f(x) = 3x^(3/6)/x^(2/3) using the quotient rule, we need to differentiate the numerator and denominator separately and then apply the quotient rule formula.
Let's start with the numerator: 3x^(3/6). The derivative of x^n (where n is a constant) is n * x^(n-1). So the derivative of 3x^(3/6) is 3 * (3/6) * x^(3/6 - 1) = (9/6) * x^(1/6).
Now let's find the derivative of the denominator: x^(2/3). Following the same rule, the derivative is (2/3) * x^(2/3 - 1) = (2/3) * x^(-1/3).
Finally, we can apply the quotient rule: (f' * g - f * g') / g^2. Substituting our derivatives into the formula, we get: ((9/6) * x^(1/6)) * (x^(2/3)) - (3x^(3/6)) * ((2/3) * x^(-1/3))) / (x^(2/3))^2. Simplifying this expression will give us the final derivative.