Results 1 to 3 of 3

Math Help - Calculus differentiation

  1. #1
    Newbie
    Joined
    Mar 2008
    Posts
    1

    Calculus differentiation

    f(x) = 3 + the integral from 8 to x cubed of f(cubed root of t) dt
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by Gorie View Post
    f(x) = 3 + the integral from 8 to x cubed of f(cubed root of t) dt
    \frac{d}{dx} \left[ \int_{8}^{x^3} f(t^{1/3}) \, dt\right]


    Let u = x^3 and use the Fundamental Theorem of Calculus and the chain rule:


    = \frac{d}{du} \left[ \int_{8}^{u} f(t^{1/3}) \, dt\right] \, \times \, \frac{du}{dx}


    = f(u^{1/3}) \, \times \, 3x^2

     = f(x) \, \times \, 3x^2 = 3x^2 \, f(x).
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Flow Master
    mr fantastic's Avatar
    Joined
    Dec 2007
    From
    Zeitgeist
    Posts
    16,948
    Thanks
    5
    Quote Originally Posted by mr fantastic View Post
    \frac{d}{dx} \left[ \int_{8}^{x^3} f(t^{1/3}) \, dt\right]


    Let u = x^3 and use the Fundamental Theorem of Calculus and the chain rule:


    = \frac{d}{du} \left[ \int_{8}^{u} f(t^{1/3}) \, dt\right] \, \times \, \frac{du}{dx}


    = f(u^{1/3}) \, \times \, 3x^2

     = f(x) \, \times \, 3x^2 = 3x^2 \, f(x).
    \frac{d}{dx} \left[ \int_{8}^{x^3} f(t^{1/3}) \, dt\right]


    Alternatively, make the substitution t^{1/3} = u \Rightarrow t = u^3. Then you have

    \frac{d}{dx} \left[ \int_{8}^{x} f(u) \, 3u^2 \, du\right]


    Now apply the Fundamental Theorem of Calculus:

    = f(x) \, 3x^2.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Calculus: Differentiation
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 27th 2010, 04:13 AM
  2. Calculus question (app. of differentiation)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: March 5th 2009, 09:05 AM
  3. Calculus Implicit Differentiation
    Posted in the Calculus Forum
    Replies: 4
    Last Post: January 28th 2009, 06:30 PM
  4. Differentiation - Calculus help!
    Posted in the Business Math Forum
    Replies: 1
    Last Post: October 13th 2008, 04:31 AM
  5. Calculus - Implicit differentiation
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 27th 2008, 04:17 AM

Search Tags


/mathhelpforum @mathhelpforum