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salantis [7]
3 years ago
10

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 su

re how to go about it. Any help would be greatly appreciated

Mathematics
1 answer:
Daniel [21]3 years ago
8 0

Answer:

\displaystyle f'(\frac{\pi}{4})=2

Step-by-step explanation:

<u>Trigonometric Derivatives</u>

We have the function

\displaystyle f(x)=\sin x \csc x-\frac{1}{\tan x}

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

\displaystyle \sin x =\frac{1}{\csc x}

Also, the reciprocal of the tangent is the cotangent:

\displaystyle \cot x =\frac{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'(\frac{\pi}{4})= \csc^2 \frac{\pi}{4}

Since csc π/4=\sqrt{2}:

\displaystyle f'(\frac{\pi}{4})= \sqrt{2}^2

\displaystyle f'(\frac{\pi}{4})=2

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