Step 1 :First, we need to find the derivative of the function $f(x)$. The function is given by $f(x) = -x^3 - 18x^2 - 81x$. Using the power rule, we find the derivative $f'(x)$:
Step 2 :f'(x) = \(-3x^2 - 36x - 81\)
Step 3 :Now, we need to find the intervals where $f''(x)$ is negative, as this indicates that $f'(x)$ is decreasing. To do this, we first find the second derivative $f''(x)$:
Step 4 :f''(x) = \(-6x - 36\)
Step 5 :Now, we need to find the intervals where $f''(x)$ is negative. To do this, we solve the inequality $f''(x) < 0$:
Step 6 :\(-6x - 36 < 0\)
Step 7 :Divide both sides by -6:
Step 8 :\(x + 6 > 0\)
Step 9 :So, the inequality is satisfied when $x > -6$. Therefore, the interval on which $f'(x)$ is decreasing is:
Step 10 :\boxed{(-6, \infty)}