Results 1 to 3 of 3

Math Help - Use The Chain Rule To Prove f+g is Differentiable

  1. #1
    Junior Member
    Joined
    Nov 2009
    From
    Pocatello, ID
    Posts
    59

    Use The Chain Rule To Prove f+g is Differentiable

    Let f,g:E\longrightarrow F (finite normed vector spaces) be differentiable functions. Use the chain rule to prove that f+g is also differentiable.

    The chain rule has to do with composite functions, I know, so I know somehow I have to think of f+g as a composite function, but I'm just not seeing it.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member Failure's Avatar
    Joined
    Jul 2009
    From
    Zürich
    Posts
    555
    Quote Originally Posted by mathematicalbagpiper View Post
    Let f,g:E\longrightarrow F (finite normed vector spaces) be differentiable functions. Use the chain rule to prove that f+g is also differentiable.

    The chain rule has to do with composite functions, I know, so I know somehow I have to think of f+g as a composite function, but I'm just not seeing it.
    Let me see: You want to represent
    f+g\!:E\ni x\mapsto f(x)+g(x)\in F
    as the composition of two "functions", right?
    What about taking
    f\otimes g\!:E\ni x\mapsto (f(x),g(x))\in F^2 and +\!: F^2\ni (x,y) \mapsto x+y\in F,
    since you have that f+g = +\circ \otimes
    Oops, that looks a bit funny, I must say. - What I mean is that f+g equals "applying + after \otimes", of course.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Bruno J.'s Avatar
    Joined
    Jun 2009
    From
    Canada
    Posts
    1,266
    Thanks
    1
    Awards
    1
    Failure has it right! (Funny thing to say )

    Remember that the map (x,y)\mapsto x+y is differentiable, being a linear transformation.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Prove d/dθ(sinθ) = pie/180cosθ (chain rule)
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 19th 2010, 02:00 PM
  2. Chain Rule Inside of Chain Rule
    Posted in the Calculus Forum
    Replies: 3
    Last Post: October 22nd 2009, 08:50 PM
  3. Replies: 5
    Last Post: October 19th 2009, 01:04 PM
  4. Replies: 3
    Last Post: May 25th 2009, 06:15 AM
  5. Replies: 2
    Last Post: December 13th 2007, 05:14 AM

Search Tags


/mathhelpforum @mathhelpforum