Results 1 to 4 of 4

Math Help - find y'(x) and y''(x) when y is defined with an integral

  1. #1
    Member
    Joined
    Feb 2011
    Posts
    75

    find y'(x) and y''(x) when y is defined with an integral

    For x > 0, define y(x) := x \int_0^{log x} \! \sqrt{1 + e^t} dt -(2/3)(1 + x)^{3/2}
    Calculate y'(x) := dy/dx
    and
    y''(x) := d^2y/dx^2
    Follow Math Help Forum on Facebook and Google+

  2. #2
    A Plied Mathematician
    Joined
    Jun 2010
    From
    CT, USA
    Posts
    6,318
    Thanks
    4
    Awards
    2
    You'll need the Leibniz Rule, the product rule, and the chain rule to differentiate your y(x). What do you get?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,361
    Thanks
    1311
    Since there is no x inside the integral, strictly speaking you don't need the full "Leibniz rule", just the "Fundamental theorem of Calculus" and the chain rule.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Oct 2008
    Posts
    1,034
    Thanks
    49
    Just in case a picture helps...



    ... where (key in spoiler) ...

    Spoiler:


    ... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case x), and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).

    Then, combining this with...



    ... the Fundamental Theorem of Calculus, we have...



    _________________________________________

    Don't integrate - balloontegrate!

    Balloon Calculus; standard integrals, derivatives and methods

    Balloon Calculus Drawing with LaTeX and Asymptote!
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Show that an integral is well defined
    Posted in the Differential Geometry Forum
    Replies: 8
    Last Post: April 21st 2010, 12:15 AM
  2. Deriving defined Integral
    Posted in the Calculus Forum
    Replies: 4
    Last Post: February 9th 2010, 07:00 AM
  3. Replies: 7
    Last Post: October 30th 2009, 12:43 AM
  4. find where the slope is defined..
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 23rd 2009, 04:52 PM
  5. Limit of defined integral
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 18th 2009, 12:58 PM

Search Tags


/mathhelpforum @mathhelpforum