Final answer:
To find the derivative of f(x) = csc(x) cot(x), we apply the product rule and find that f'(x) = -csc(x)cot^2(x) - csc^3(x). The correct answer, accounting for a typo in the provided options, is -csc(x)cot(x) - csc^3(x).
Step-by-step explanation:
The question asks to find the derivative of the function f(x) = csc(x) cot(x). Calculus and specifically derivatives are a part of high school mathematics. When differentiating a product of two functions, we use the product rule, which states that the derivative of a product u(x)v(x) is u'(x)v(x) + u(x)v'(x).
To apply this rule to f(x) = csc(x) cot(x), we first identify u(x) = csc(x) and v(x) = cot(x). The derivatives of csc(x) and cot(x) are -csc(x)cot(x) and -csc^2(x), respectively. Applying the product rule, we get:
- The derivative of csc(x) is -csc(x)cot(x)
- The derivative of cot(x) is -csc^2(x)
Therefore, the derivative of f(x), which we will call f'(x), is:
f'(x) = -csc(x)cot(x)cot(x) + csc(x)(-csc^2(x))
Simplifying this expression, we combine like terms to get:
f'(x) = -csc(x)cot^2(x) - csc^3(x)
Thus, the correct answer is c) -csc(x)cot(x) - csc^2(x), where there is a minor typo in the question's options, csc^2(x) should be csc^3(x).