Final answer:
To find the derivative of the function f(x) = 6th root of x, we rewrite it as f(x) = x^(1/6) and apply the power rule to obtain the derivative, f'(x) = (1/6)x^(-5/6). The correct answer is A. (1/6) * x^(-5/6). The correct option is A.
Step-by-step explanation:
The derivative of f(x) = 6th root of x can be found using the power rule. The power rule states that if f(x) = x^n, then the derivative of f(x) is f'(x) = nx^(n-1). In this case, the power is 1/6, so the derivative is f'(x) = (1/6)x^(1/6 - 1) = (1/6)x^(-5/6). Therefore, the correct answer is option A: (1/6)x^(-5/6).
The question asks us to find the derivative of the function f(x) = 6th root of x. This can be rewritten using exponents as f(x) = x^(1/6). The rule for differentiating a power of x is to multiply by the exponent and then subtract one from the exponent. So, applying the power rule:
f'(x) = (1/6)x^(1/6 - 1)
Which simplifies to:
f'(x) = (1/6)x^(-5/6)
Therefore, the correct answer is A. (1/6) * x^(-5/6).