00:01
We are looking for f prime of x, which is the derivative of f of x.
00:05
So it's the derivative of this function.
00:21
So notice we have one function divided by another function, so we can use caution now, which tells us this is equal to the derivative of the top, and that's multiplied by the bottom.
00:36
Then we subtract and take the top, multiplied by the derivative of the bottom.
00:43
And that's all over the bottom squared.
00:55
So first we will input those derivatives.
00:59
Derivative of cotangent is negative cosecant squared.
01:10
And then the derivative of 1 plus co -secant x, the one can be ignored because we're taking the derivative into constant.
01:18
So we have negative co -secant x, cotangent x.
01:25
And the denominator can stay the same.
01:31
Now, it'll be helpful to take a common factor out of this expression, and that common factor would be co -secent x.
01:46
So, oh, and firstly, of course, these two negatives cancel out, so i'll cross one of the math, the other will be coming up to us, and we will factor out co -secent.
01:58
So we have co -secent x, then we have from the first term, negative co -secent x times 1 plus cosecant x and then in the next part we have plus cotangent times cotangent which is just cotangent squared and the denominator can stay the same so this is um this is equal to the derivative however we can of course simplify this top more so um firstly it'll be helpful to expand these brackets so we have, um, yeah, we have as follows.
02:50
First, negative co -secant x and then minus co -secent squared x.
02:59
And then we can keep that cotangent squared x.
03:11
Now, remember, it can be derived from the pythagorean identity that co -secent, co -secret squared is equal to 1 plus cotangent squared.
03:30
And we're going to use that...