# Thread:

1. ## Differentiation in Banach Space

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

Find a derivative (Fréchet derivative) of the operator
$F:C[0,\pi]\to C[0,\pi], F(u)=\sin u(x),$
at the point $u_0(x) = \cos(x).$

There is an answer to this problem in the book: $F'(u_0)=\cos\sin x$, 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 $\cos\sin x$ 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 $F(u)=\cos u(x), u_0(x) = \sin(x)$ with an answer $\sin\cos x$ in this book, so I think this is not any mistake or something. Can anyone please help with this?

Thanks in advance.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 6, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9674002528190613, "perplexity_flag": "head"}