Question

Let f be the function given by f(x) =cosx-cscx . what is the value of f'(1)? A) f ′(1)=−sin(1)

B) f ′(1)=sin(1)
C) f ′(1)=cot(1)
D) f ′(1)=sin(1)+cot(1)

Answer 1

To find the value of f'(1) for the function f(x) = cos(x) - csc(x), we need to first find the derivative of the function and evaluate it at x = 1. The derivative of f(x) is -sin(x) + csc(x)cot(x). Substituting x = 1, we find that f'(1) = -sin(1) + cos(1)/sin^2(1). Therefore, the correct answer is A) f ′(1)=−sin(1).

Step-by-step explanation:

To find the derivative of the function f(x) = cos(x) - csc(x) and evaluate f'(1), we can use the rules of differentiation. Firstly, we can use the power rule to find the derivative of cos(x) and csc(x). The derivative of cos(x) is -sin(x) and the derivative of csc(x) is -csc(x)cot(x). Therefore, the derivative of f(x) = cos(x) - csc(x) is f'(x) = -sin(x) + csc(x)cot(x).

To evaluate f'(1), we substitute x = 1 into the derivative function: f'(1) = -sin(1) + csc(1)cot(1). However, csc(1) = 1/sin(1) and cot(1) = cos(1)/sin(1). Substituting these values, f'(1) = -sin(1) + (1/sin(1))(cos(1)/sin(1)). Simplifying further, f'(1) = -sin(1) + cos(1)/sin^2(1).

Therefore, the value of f'(1) is -sin(1) + cos(1)/sin^2(1), which is option A) f ′(1)=−sin(1).