Question

(problem 83)

Any help would be appreciated :) I have the answer key, I just don't know how to get the answer, so explaining each step would be great!

Answer 1

To find the derivative of this function, there is a property that we should know called the Constant Multiple Rule, which says:

`(d)/(dx)[cf(x)] = cf'(x)` (where
`c` is a constant)

Remember that the derivative of
`\csc(x)` is
`-\csc(x)\cot(x)`. However, you may notice that we are finding the derivative of
`(1)/(2)\csc(2x)`, not
`(1)/(2) \csc(x)`. So, we are going to have to use the chain rule. To complete the chain rule for the derivative of a trigonometric function (in layman's terms) is basically the following: First, complete the derivative of the trig function as you would if what was inside the trig function is
`x`. Then, take the derivative of what's inside of the trig function and multiply it by what you found in the first step.

Let's apply that to our problem. Right now, I am not going to worry about the
`(1)/(2)` at the front of the equation, since we can just multiply it back in at the end of our problem. So, let's examine
`\csc(2x)`. We see that what's inside the trig function is
`2x`, which has a derivative of 2. Thus, let's first find the derivative of
`\csc(2x)` as if
`2x` was just
`x` and then multiply it by 2.

The derivative of
`\csc(2x)` would first be
`-\cot(2x)\csc(2x)`. Multiplying it by 2, we get our derivative of
`-2\cot(2x)\csc(2x)`. However, don't forget to multiply it by the
`(1)/(2)` that we removed near the beginning. This gives us our final derivative of
`-\cot(2x)\csc(2x)`.

Remember that we now have to find the derivative at the given point. To do this, simply "plug in" the point into the derivative using the x-coordinate. This is shown below:

`-\cot[2((\pi)/(4))]\csc[2((\pi)/(4))]`

`-\cot((\pi)/(2))\csc((\pi)/(2))`

`-(0)(1) = \boxed{0}`

Our final answer is 0.