To find the derivative of the function:

$f(x) = \csc^3(9x - 5)$

we'll apply the chain rule and product rule, as this function involves a composition of functions.

Steps:

1. Outer function: The outer function is $\csc^3(u)$, where $u = 9x - 5$.

   • The derivative of $\csc^3(u)$ with respect to $u$ is found using the chain rule: $\frac{d}{du}[\csc^3(u)] = 3\csc^2(u) \cdot (-\csc(u) \cot(u)) = -3\csc^3(u)\cot(u)$

2. Inner function: The inner function is $u = 9x - 5$, and its derivative is: $\frac{d}{dx}(9x - 5) = 9$

Putting it together:

Using the chain rule: