Results 1 to 3 of 3
Like Tree2Thanks
  • 1 Post By Soroban
  • 1 Post By Prove It

Math Help - Simplifying a Derivative

  1. #1
    Newbie
    Joined
    Jun 2012
    From
    Canada
    Posts
    20

    Question Simplifying a Derivative

    I'm having an issue with the following derivative.

    y=\frac{x^3-3x^2+2\sqrt{x}-5}{\sqrt{x}}

    I use the product rule and get up to this point:

    =\frac{3x^2-6x+x^{-1/2}}{\sqrt{x}}-\frac{x^3-3x^2+2x^{1/2}-5}{2\sqrt{x^3}}

    Wolfram Alpha tells me the answer is:

    =\frac{5x^3-9x^2+5}{2\sqrt{x^3}}

    My fraction skills are not the best-- far from best. I tried multiplying the left side by \frac{2\sqrt{x}}{2\sqrt{x}} but that made a big mess. Maybe it was the right thing to do but I just made a mess and couldn't see how to make it work. Thanks for any help. I've recently returned to school after a 17 year hiatus.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,735
    Thanks
    642

    Re: Simplifying a Derivative

    Hello, Maskawisewin!

    \text{Differentiate: }\:y\:=\:\frac{x^3-3x^2+2\sqrt{x}-5}{\sqrt{x}}

    I would not avoid the Quotient Formula by always changing a quotient to a product
    . . unless you are very good at handling negative and fractional exponents.


    Besides, I would handle it this way:

    y \;=\;\frac{x^3 - 3x^2 + 2x^{\frac{1}{2}} - 5}{x^{\frac{1}{2}}} \;=\;\frac{x^3}{x^{\frac{1}{2}}} - \frac{3x^2}{x^{\frac{1}{2}}} + \frac{2x^{\frac{1}{2}}}{x^{\frac{1}{2}}} - \frac{5}{x^{\frac{1}{2}}} \;=\; x^{\frac{5}{2}}\: \:- 3x^{\frac{3}{2}} \:+\: 2 \:-\: 5x^{-\frac{1}{2}}

    Then: . y' \;=\;\tfrac{5}{2}x^{\frac{3}{2}} - \tfrac{9}{2}x^{\frac{1}{2}} + \tfrac{5}{2}x^{-\frac{3}{2}} \;=\;\tfrac{1}{2}x^{-\frac{3}{2}}\left(5x^3 - 9x^2 + 5\right)

    Therefore: . y' \;=\;\frac{5x^3 - 9x^2 + 5}{2x^{\frac{3}{2}}}
    Thanks from Maskawisewin
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    11,547
    Thanks
    1418

    Re: Simplifying a Derivative

    Quote Originally Posted by Maskawisewin View Post
    I'm having an issue with the following derivative.

    y=\frac{x^3-3x^2+2\sqrt{x}-5}{\sqrt{x}}

    I use the product rule and get up to this point:

    =\frac{3x^2-6x+x^{-1/2}}{\sqrt{x}}-\frac{x^3-3x^2+2x^{1/2}-5}{2\sqrt{x^3}}

    Wolfram Alpha tells me the answer is:

    =\frac{5x^3-9x^2+5}{2\sqrt{x^3}}

    My fraction skills are not the best-- far from best. I tried multiplying the left side by \frac{2\sqrt{x}}{2\sqrt{x}} but that made a big mess. Maybe it was the right thing to do but I just made a mess and couldn't see how to make it work. Thanks for any help. I've recently returned to school after a 17 year hiatus.
    To get a common denominator, note that \displaystyle \begin{align*} 2\sqrt{x^3} = 2\sqrt{x\cdot x^2} = 2\sqrt{x}{\sqrt{x^2}} = 2x\sqrt{x} \end{align*}, so you will need to multiply the top and bottom of the first term by \displaystyle \begin{align*} 2x \end{align*}, not \displaystyle \begin{align*} 2\sqrt{x} \end{align*}.
    Thanks from tricxta
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Help Simplifying a 2nd Derivative
    Posted in the Calculus Forum
    Replies: 6
    Last Post: November 27th 2009, 03:10 PM
  2. Simplifying a Derivative
    Posted in the Calculus Forum
    Replies: 2
    Last Post: November 17th 2009, 04:36 PM
  3. Simplifying a Derivative
    Posted in the Calculus Forum
    Replies: 4
    Last Post: September 16th 2009, 07:46 PM
  4. simplifying a derivative
    Posted in the Calculus Forum
    Replies: 4
    Last Post: June 13th 2009, 05:22 PM
  5. Further Simplifying A Derivative
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 8th 2009, 04:57 AM

Search Tags


/mathhelpforum @mathhelpforum