Final answer:
The derivative of the function f(x) = (-x^5 - 4x^2 - 3x) sin(x) is found by applying the product rule and is f'(x) = (-5x^4 - 8x - 3)sin(x) + (-x^5 - 4x^2 - 3x)cos(x).
Step-by-step explanation:
To find the derivative of f(x) = (-x^5 - 4x^2 - 3x) sin(x), we need to apply the product rule which states that if we have two functions u(x) and v(x), the derivative of their product u(x)v(x) is u'(x)v(x) + u(x)v'(x).
In this case, u(x) = -x^5 - 4x^2 - 3x and v(x) = sin(x). First, we find the derivative of u(x), which is u'(x) = -5x^4 - 8x - 3. The derivative of v(x) is v'(x) = cos(x).
Applying the product rule:
f'(x) = (-5x^4 - 8x - 3)sin(x) + (-x^5 - 4x^2 - 3x)cos(x)
Which is the derivative f'(x) that we were looking for.