Final answer:
The derivative of f(x) = -xcos3x is found using the product rule and chain rule, resulting in f'(x) = -cos3x + 3xsin3x.
Step-by-step explanation:
To find the derivative of the function f(x) = -xcos3x, we will use the product rule.
The product rule states that when you have a function that is the product of two functions, its derivative is found by taking the derivative of each and multiplying each by the other function.
So if we have two functions u(x) and v(x), the derivative of u(x)v(x) is u'(x)v(x) + u(x)v'(x).
Using the product rule, let's differentiate f(x) = -xcos3x:
First, let u(x) = -x. The derivative of u with respect to x, u'(x), is -1.
Next, let v(x) = cos3x.
The derivative of v with respect to x, v'(x), is -3sin3x (using the chain rule).
Now, apply the product rule: f'(x) = u'(x)v(x) + u(x)v'(x).
Plugging in the derivatives, we get f'(x) = (-1)cos3x + (-x)(-3sin3x).
Therefore, the derivative of f(x) is f'(x) = -cos3x + 3xsin3x.