Results 1 to 7 of 7
Like Tree2Thanks
  • 1 Post By HallsofIvy
  • 1 Post By dennydengler

Math Help - What to do next? (derivative)

  1. #1
    Newbie
    Joined
    Aug 2014
    From
    N/A
    Posts
    3

    What to do next? (derivative)

    Hi, I'm beginner to calculus and I forgot my algebra because I stop my study for years. Now I don't know what next after I plug the value to the formula of quotient rule in derivate.

    This is the given.
    y = \frac{x^3}{(x^2+4)^2}

    Then I think this is the derivative of the given function, but how to solve or factor or simplify this?
    = \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}

    If you can show me the full solution then explain what process did you do, it will help me alot. Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2013
    From
    California
    Posts
    2,462
    Thanks
    950

    Re: What to do next? (derivative)

    Quote Originally Posted by avahdon View Post
    Hi, I'm beginner to calculus and I forgot my algebra because I stop my study for years. Now I don't know what next after I plug the value to the formula of quotient rule in derivate.

    This is the given.
    y = \frac{x^3}{(x^2+4)^2}

    Then I think this is the derivative of the given function, but how to solve or factor or simplify this?
    = \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}

    If you can show me the full solution then explain what process did you do, it will help me alot. Thanks
    if

    $f(x) = \dfrac{g(x)}{h(x)}$

    and

    $f^\prime(x)=\dfrac{df}{dx}(x)$

    then

    $f^\prime(x)=\dfrac{g^\prime(x)h(x)-g(x)h^\prime(x)}{(h(x))^2}$

    so in this case

    $g(x)=x^3$

    $g^\prime(x)=3x^2$

    $h(x)=(x^2+4)^2$

    $h^\prime(x)=2(x^2+4)2x$

    you should be able to put all the pieces together from here.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2014
    From
    N/A
    Posts
    3

    Re: What to do next? (derivative)

    Quote Originally Posted by romsek View Post
    if

    $f(x) = \dfrac{g(x)}{h(x)}$

    and

    $f^\prime(x)=\dfrac{df}{dx}(x)$

    then

    $f^\prime(x)=\dfrac{g^\prime(x)h(x)-g(x)h^\prime(x)}{(h(x))^2}$

    so in this case

    $g(x)=x^3$

    $g^\prime(x)=3x^2$

    $h(x)=(x^2+4)^2$

    $h^\prime(x)=2(x^2+4)2x$

    you should be able to put all the pieces together from here.
    I already solve the derivative and I have this = \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}

    my question now is how to simplify, solve, or factor this? because my teacher don't accept this answer
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,693
    Thanks
    1466

    Re: What to do next? (derivative)

    Quote Originally Posted by avahdon View Post
    Hi, I'm beginner to calculus and I forgot my algebra because I stop my study for years. Now I don't know what next after I plug the value to the formula of quotient rule in derivate.

    This is the given.
    y = \frac{x^3}{(x^2+4)^2}

    Then I think this is the derivative of the given function, but how to solve or factor or simplify this?
    = \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}

    If you can show me the full solution then explain what process did you do, it will help me alot. Thanks
    One thing you should be able to see that there is a term of the form x^2+ 4 in both terms in the numerator and one in the denominator so you can cancel those:
    \frac{(x^2+ 4)(3x^2)- (x^3)(2x)}{(x^2+ 4)^3}
    (You did recognize that ((x^2+ 4)^2)^2 is (x^2+ 4)^4, right?)

    Now multiply out (x^2+ 4)(3x^3) and (x^3)(2x) and subtract.

    (I strongly recommend that you start reviewing algebra and trig.)
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,738
    Thanks
    644

    Re: What to do next? (derivative)

    Hello, avahdon!

    y \:=\: \frac{x^3}{(x^2+4)^2}

    . . y' \:=\: \frac{(x^2+4)^2(3x^2)-(x^3)2(x^2+4)(2x)}{((x^2+4)^2)^2}

    How to simplify this?

    You have: . y' \;=\;\frac{3x^2(x^2+4)^2 - 4x^4(x^2+4)}{(x^2+4)^4}

    Factor: . y' \;=\;\frac{x^2(x^2+4)\big[3(x^2+4) - 4x^2\big]}{(x^2+4)^4}

    Reduce: . y' \;=\;\frac{x^2\big[3x^2+12 - 4x^2\big]}{(x^2+4)^3}

    Simplify: . y' \;=\;\frac{x^2(12-x^2)}{(x^2+4)^3}
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Aug 2014
    From
    N/A
    Posts
    3

    Re: What to do next? (derivative)

    Quote Originally Posted by HallsofIvy View Post
    One thing you should be able to see that there is a term of the form x^2+ 4 in both terms in the numerator and one in the denominator so you can cancel those:
    \frac{(x^2+ 4)(3x^2)- (x^3)(2x)}{(x^2+ 4)^3}
    (You did recognize that ((x^2+ 4)^2)^2 is (x^2+ 4)^4, right?)

    Now multiply out (x^2+ 4)(3x^3) and (x^3)(2x) and subtract.

    (I strongly recommend that you start reviewing algebra and trig.)
    thanks, can you tell me which topic should i review in algebra and trig?

    Quote Originally Posted by Soroban View Post
    Hello, avahdon!


    You have: . y' \;=\;\frac{3x^2(x^2+4)^2 - 4x^4(x^2+4)}{(x^2+4)^4}

    Factor: . y' \;=\;\frac{x^2(x^2+4)\big[3(x^2+4) - 4x^2\big]}{(x^2+4)^4}

    Reduce: . y' \;=\;\frac{x^2\big[3x^2+12 - 4x^2\big]}{(x^2+4)^3}

    Simplify: . y' \;=\;\frac{x^2(12-x^2)}{(x^2+4)^3}
    thanks.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Jul 2014
    From
    Waterloo, ON
    Posts
    82
    Thanks
    18

    Re: What to do next? (derivative)

    Alternatively, use logarithmic differentiation.
    y=x^3/(x^2+4)^2, so
    ln(y) = ln(x^3) - ln(x^2+4)^2 = 3ln(x)-2ln(x^2+4)
    Differentiating both sides, you have
    y'/y = 3/x - 2*1/(x^2+4)*2x = 3/x -4x/(x^2+4) = [3(x^2+4)-4x^2]/[x(x^2+4)] = [-x+12]/[x(x^2+4)]
    Now, solving for y', you just multiply both sides by y:
    y' = [-x+12]/[x(x^2+4)] * x^3/(x^2+4)^2 = (12-x)/(x^2+4) * x^2/(x^2+4)^2 = (12x^2-x^3)/(x^2+4)^3. //
    Thanks from topsquark
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. difference between intrinsic derivative and covariant derivative
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 16th 2013, 01:58 AM
  2. Replies: 4
    Last Post: October 17th 2012, 06:51 AM
  3. contuous weak derivative $\Rightarrow$ classic derivative ?
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: April 22nd 2011, 02:37 AM
  4. Replies: 0
    Last Post: January 24th 2011, 11:40 AM
  5. Replies: 2
    Last Post: November 6th 2009, 02:51 PM

Search Tags


/mathhelpforum @mathhelpforum