Thread:

1. Confusion about Total Derivative

Hello everyone, I was a little confused about something and hope that someone can help me.

Let $U \subseteq \mathbb{R}$ be open, and if a function $f : U \rightarrow \mathbb{R}$ is differentiable on $U,$ then for each $u\in U$ there must exist a linear map $Df(u):\mathbb{R} \rightarrow \mathbb{R}$ such that

$\displaystyle\lim_{h\rightarrow 0} \displaystyle\frac{| f(u+h)-f(u) - Df(u)h|}{|h|} = 0$

Then this is true if and only if $Df(u) = f'(u) = \lim_{h \rightarrow 0} \displaystyle\frac{f(u+h)-f(u)}{h}$. So doesn't this mean that if we can differentiate f, it's then differentiable as a linear map? I know that sounds stupid, but I'm trying to think of the result that if the partial derivatives of f are continuous, then f is differentiable in this special case. So is continuity of the derivative not important in the real case?

What about parametric equations, for example $f: \mathbb{R} \rightarrow \mathbb{R}^2$ where $f(x) = (y_1(x), y_2(x))$. On wikipedia it just says that the derivative of f is $f'(x) = (y_1'(x), y_2'(x)).$ But do the derivatives of $y_1, y_2$ only have to exist for f to be differentiable? Does it matter if they're not continuous?

Thanks for any help

2. Re: Confusion about Total Derivative

Sorry, I've discovered that this is a known result:

If $\gamma:\mathbb{R} \rightarrow \mathbb{R}^m$ is differentiable if and only if its component functions are differentiable in the sense of single-variable calculus.
{"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": 12, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "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": 0.9564386010169983, "perplexity_flag": "head"}