00:01
Buckle your seatbelt for this very long response.
00:04
So they have the natural log of cosine squared of this tangent function, inverse tangent, excuse me, of natural log of the absolute value of x.
00:19
And let's get these brackets closed, the brace close, and then the absolute value.
00:25
All right.
00:26
So let me try and write small so i can fit this.
00:29
Because the chain rule says take the derivative of natural log first, which is one over that piece, which would be leaving cosine squared of inverse tangent, of natural log of x alone.
00:47
But then you have to multiply by the next big thing is, let's circle this in blue, is the two.
00:53
So i have to take the derivative of all the inside, but you have to worry about cosine squared first.
00:58
So bring that in front, leaving cosine of all of that alone.
01:12
And then times the next big thing, i can actually leave it in the numerator, but i'm just going to move over here, the derivative of cosine, which would be negative sign of leaving everything else alone, of the inverse tangent, of natural log of x.
01:32
And now we have to do the derivative of inverse tangent, which is one, over 1 plus the piece squared and the piece that we're talking about is natural log of x.
01:55
But then we have to take the derivative of natural log of x, which would be 1 over x.
02:03
Now, i know it says to simplify the results, but i'm not sure what else there would be to simplify.
02:12
Aside from maybe, this is just a big maybe, you could cancel out one.
02:17
Of these cosines and then you have a i don't even know negative sign over cosine would be the same thing as negative tangent but if you have negative tangent with inverse tangent i don't know if they would let you cancel those two things out so you'd be left with natural log of x over x times that one plus natural log of x squared.
02:57
I think that's all you can do to simplify...