Final Answer:
To find the derivative of f(x) = tan(3x) * cos(x), apply the product rule. The result is f'(x) = 3tan²(3x)sec²(3x) - sin(x)tan(3x).
Step-by-step explanation:
The derivative of a product of two functions can be found using the product rule, which states that for two functions u(x) and v(x), (uv)' = u'v + uv'. Applying this rule to f(x) = tan(3x) * cos(x), let u(x) = tan(3x) and v(x) = cos(x).
The derivative u'(x) is found using the chain rule for tan(3x), resulting in 3tan²(3x)sec²(3x). The derivative v'(x) is simply -sin(x).
Combine these results using the product rule, and simplify to obtain f'(x) = 3tan²(3x)sec²(3x) - sin(x)tan(3x). This is the final answer for the derivative of f(x).