Given the function f(x) and its derivative f'(x), let's introduce a second function g(x) such that:
g(x) = (f(x) + 3x⁴ - 3x - 1)⁵
The question is asking for the value of the derivative of g(x) evaluated at x = -2.
Let's start with the substitution of the given function f(x) into g(x):
g(x) = (-11 + 3x⁴ - 3x - 1)⁵ = (3x⁴ - 3x - 12)⁵
As we can see, the original function f(x) simplifies to a constant, -11, after replaced by the given f(x).
Now, we will calculate the derivative of g(x).
By using the chain rule, which states that the derivative of a composition of functions is the derivative of the outer function times the derivative of the inner function, we can find the derivative of g(x).
The outer function in our case is h(u) = u⁵ and its derivative is h'(u) = 5u⁴.
The inner function is w(x) = 3x⁴ - 3x - 12 and its derivative is w'(x) = 12x³ - 3.
Therefore, the derivative g'(x) of the function g(x) is:
g'(x) = h'(u) * w'(x) = 5*(3x⁴ - 3x - 12)⁴ * (12x³ - 3)
= (60x³ - 15)*(3x⁴ - 3x - 12)⁴
Finally, to calculate the value of g'(x) at x = -2, we substitute x = -2 into g'(x):
g'(-2) = (60*(-2)³ - 15)*(3*(-2)⁴ - 3*(-2) - 12)⁴
= -1540289520
So, the value of the derivative of g(x) at x = -2 is -1540289520.