Results 1 to 4 of 4

Math Help - Derivatives of natual logs

  1. #1
    Junior Member
    Joined
    Aug 2008
    Posts
    52

    Derivatives of natual logs

     y= \ln (x \sqrt {x^2-1})
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Dec 2008
    From
    Scotland
    Posts
    901
    Quote Originally Posted by yeloc View Post
     y= \ln (x \sqrt {x^2-1})
     \frac{d}{dx} \ln(f(x)) = \frac{1}{f(x)} \times \frac{d}{dx} (f(x))

    Hence:

     \frac{d}{dx} \ln(x\sqrt{x^2-1}) = \frac{1}{x\sqrt{x^2-1}} \times \frac{d}{dx} (x\sqrt{x^2-1})

      = \frac{1}{x\sqrt{x^2-1}} \times \frac{d}{dx} (\sqrt{x^4-x^2})

      = \frac{1}{x\sqrt{x^2-1}} \times \frac{d}{dx} ({x^4-x^2})^{\frac{1}{2}}

    Now use the chain rule again:

      = \frac{1}{x\sqrt{x^2-1}} \times \frac{1}{2}(x^4-x^2)^{\frac{-1}{2}} \times \frac{d}{dx} (x^4-x^2)

      = \frac{1}{x\sqrt{x^2-1}} \times \frac{1}{2\sqrt{x^4-x^2}} \times \frac{d}{dx} (x^4-x^2)

      = \frac{1}{x\sqrt{x^2-1}} \times \frac{1}{2x\sqrt{x^2-1}} \times \frac{d}{dx} (x^4-x^2)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Aug 2008
    Posts
    52
    Thanks! I found my mistake; I wasn't multiplying by the inside function when doing the chain rule.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Math Engineering Student
    Krizalid's Avatar
    Joined
    Mar 2007
    From
    Santiago, Chile
    Posts
    3,654
    Thanks
    13
    f(x)=\ln x+\frac{1}{2}\ln \left( {{x}^{2}}-1 \right)\implies f'(x)=\frac{1}{x}+\frac{x}{{{x}^{2}}-1}, and we're done.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Derivatives with Trig and Natural Logs
    Posted in the Calculus Forum
    Replies: 2
    Last Post: October 19th 2011, 05:00 PM
  2. derivatives with natural logs
    Posted in the Calculus Forum
    Replies: 3
    Last Post: May 4th 2010, 01:02 PM
  3. derivatives using natural logs
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 30th 2009, 08:55 PM
  4. Third Law of Logs Proof (using derivatives)
    Posted in the Calculus Forum
    Replies: 1
    Last Post: March 1st 2009, 06:15 PM
  5. Derivatives of Natural Logs question
    Posted in the Calculus Forum
    Replies: 3
    Last Post: July 25th 2007, 03:13 PM

Search Tags


/mathhelpforum @mathhelpforum