Question

Normally I don’t ask for help but I’m very lost on question 4. I know I need to first find the derivative of f(x) but I’m not sure how to go about it. Any help would be greatly appreciated

Answer 1

`\displaystyle f'((\pi)/(4))=2`

Explanation:

Trigonometric Derivatives

We have the function

`\displaystyle f(x)=\sin x \csc x-(1)/(\tan x)`

We must recall that the sine and the cosecant are reciprocal functions, i.e.:

`\displaystyle \sin x =(1)/(\csc x)`

Also, the reciprocal of the tangent is the cotangent:

`\displaystyle \cot x =(1)/(\tan x)`

Thus:

`\displaystyle f(x)=1-\cot x`

Now it's easier to take the derivative

`\displaystyle f'(x)=0+\csc^2 x= \csc^2 x`

Evaluating for x=π/4:

`\displaystyle f'((\pi)/(4))= \csc^2 (\pi)/(4)`

Since csc π/4=
`√(2)`:

`\displaystyle f'((\pi)/(4))= √(2)^2`

`\displaystyle f'((\pi)/(4))=2`