Final answer:
The derivative of f(t) = 3^(3t)/t can be found using the product and chain rules, resulting in f'(t) = 3^(3t) × (3ln(3)/t - 1/t^2).
Step-by-step explanation:
The derivative of f(t) = 3^(3t)/t requires the use of both the product rule and the chain rule for derivatives, as the function can be seen as the product of two functions, u(t) = 3^(3t) and v(t) = 1/t. To find the derivative f'(t), use the product rule: f'(t) = u'(t) × v(t) + u(t) × v'(t). The derivative of u(t) with respect to t, u'(t), can be found by using the chain rule, which in the case of an exponential function with a base other than e, is u'(t) = 3^(3t) × ln(3) × 3, and the derivative of v(t) is v'(t) = -1/t^2. Therefore, f'(t) = 3^(3t) × ln(3) × 3 × (1/t) + 3^(3t)/t^2 × (-1), or simply f'(t) = 3^(3t) × (3ln(3)/t - 1/t^2).