Results 1 to 2 of 2

Thread: Confusion about Total Derivative

  1. #1
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347

    Confusion about Total Derivative

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

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

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

    Then this is true if and only if $\displaystyle Df(u) = f'(u) = \lim_{h \rightarrow 0} \displaystyle\frac{f(u+h)-f(u)}{h}$. 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 $\displaystyle f: \mathbb{R} \rightarrow \mathbb{R}^2$ where $\displaystyle f(x) = (y_1(x), y_2(x))$. On wikipedia it just says that the derivative of f is $\displaystyle f'(x) = (y_1'(x), y_2'(x)).$ But do the derivatives of $\displaystyle y_1, y_2$ only have to exist for f to be differentiable? Does it matter if they're not continuous?

    Thanks for any help
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member slevvio's Avatar
    Joined
    Oct 2007
    Posts
    347

    Re: Confusion about Total Derivative

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

    If $\displaystyle \gamma:\mathbb{R} \rightarrow \mathbb{R}^m$ is differentiable if and only if its component functions are differentiable in the sense of single-variable calculus.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Derivative confusion
    Posted in the Differential Equations Forum
    Replies: 6
    Last Post: Apr 22nd 2010, 08:49 PM
  2. Total derivative of f(A)=A^-1 at I
    Posted in the Calculus Forum
    Replies: 1
    Last Post: Jan 5th 2010, 11:22 PM
  3. Derivative confusion
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Apr 25th 2009, 07:35 PM
  4. Total derivative
    Posted in the Calculus Forum
    Replies: 2
    Last Post: Mar 16th 2009, 11:52 AM
  5. Total Derivative vs. Directional Derivative
    Posted in the Advanced Math Topics Forum
    Replies: 5
    Last Post: May 30th 2008, 08:42 AM

Search Tags


/mathhelpforum @mathhelpforum