Final answer:
The derivative of the function f(x) = 4 csc x is f'(x) = -4 csc x cot x, which can be simplified to f'(x) = -4 cos x/sin^2 x.
Step-by-step explanation:
To find the derivative of the function f(x) = 4 csc x, we first recognize that csc x is the reciprocal of sin x, meaning csc x = 1/sin x.
The derivative of csc x can be found using the quotient rule or by directly using the derivative identity for csc x.
The derivative of csc x with respect to x is -csc x cot x, and since we need to find the derivative of 4 csc x, we can simply multiply this derivative by 4.
Therefore, the derivative of f(x) = 4 csc x with respect to x, denoted as f'(x), is given by:
f'(x) = -4 csc x cot x
This can also be written as f'(x) = -4 (1/sin x) (cos x/sin x), which simplifies to f'(x) = -4 cos x/sin^2 x.