Results 1 to 5 of 5

Math Help - Need derivative of y = cos (x^sqrt(x))

  1. #1
    Newbie
    Joined
    Aug 2009
    Posts
    11

    Need derivative of y = cos (x^sqrt(x))

    I'm trying do it using log differentiation but I'm not really sure how to use log rules to bring down the sqrt (x) . Can I do ln y = sqrt(x) ln cos x ? It seems like I'm wrongly taking the sqrt (x) out of the bracket if I do that?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,885
    Thanks
    680
    Quote Originally Posted by potatoes View Post
    I'm trying do it using log differentiation but I'm not really sure how to use log rules to bring down the sqrt (x) . Can I do ln y = sqrt(x) ln cos x ? It seems like I'm wrongly taking the sqrt (x) out of the bracket if I do that?
    do this ...

    y = \cos(u)

    \frac{dy}{dx} = -sin(u) \cdot \frac{du}{dx}

    let u = x^{\sqrt{x}} and use logarithmic differentiation to find \frac{du}{dx}
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Aug 2009
    Posts
    11
    Ok thanks.

    So that gives me dy/dx = -sin u * (sqrt(x)lnx + 2sqrt(x)/2x) * u

    At this point does the derivative become dy/du? and then I can just leave it like that? or do I sub in x^sqrt(x) for u?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    skeeter's Avatar
    Joined
    Jun 2008
    From
    North Texas
    Posts
    11,885
    Thanks
    680
    u = x^{\sqrt{x}}

    \ln{u} = \sqrt{x} \cdot \ln{x}

    \frac{u'}{u} = \frac{1}{\sqrt{x}} + \frac{\ln{x}}{2\sqrt{x}} = \frac{2+\ln{x}}{2\sqrt{x}}

    u' = x^{\sqrt{x}} \cdot \frac{2+\ln{x}}{2\sqrt{x}}

    finally ...

    \frac{dy}{dx} = -\sin\left(x^{\sqrt{x}}\right) \cdot x^{\sqrt{x}} \cdot \frac{2+\ln{x}}{2\sqrt{x}}
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Aug 2009
    Posts
    11
    Ok great. Thanks so much.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Derivative of f(x) = x + sqrt(x)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: January 31st 2011, 09:50 PM
  2. Derivative of x sqrt(6x - 1)
    Posted in the Calculus Forum
    Replies: 2
    Last Post: September 5th 2010, 05:23 AM
  3. Replies: 2
    Last Post: September 1st 2010, 02:29 PM
  4. Replies: 2
    Last Post: November 6th 2009, 02:51 PM
  5. Replies: 3
    Last Post: October 14th 2006, 07:09 AM

Search Tags


/mathhelpforum @mathhelpforum