Differentiation in Banach Space

Hello everyone, I am having some troubles with the following problem:

Find a derivative (Fréchet derivative) of the operator

at the point

There is an answer to this problem in the book: , but I can't get it: I tried using mean value theorem, Taylor series and different trigonometry formulas to find the linear part of the increment, but I still don't understand how can we get an answer here. I also don't thinks its necessary to find Gateaux derivative here, which is equal to Frechet derivative in this case. There is also a similar problem with with an answer in this book, so I think this is not any mistake or something. Can anyone please help with this?

Thanks in advance.