Results 1 to 6 of 6

Math Help - Differentiation of Logarithmic and Exponential Functions

  1. #1
    Junior Member
    Joined
    Feb 2008
    Posts
    61

    Differentiation of Logarithmic and Exponential Functions

    Hello, I am having trouble differentiating this problem, it looks like the quotient rule is being used within the product rule, but I don't know how to combine the two of them.

    f(x)= ln(x+1/x-1) I started using the product rule but then I got lost when I had to differentiate the second term, because this would be a quotient rule in this term. If someone could illustrate how to differentiate this, Thanks...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Forum Admin topsquark's Avatar
    Joined
    Jan 2006
    From
    Wellsville, NY
    Posts
    10,211
    Thanks
    419
    Awards
    1
    Quote Originally Posted by kdogg121 View Post
    Hello, I am having trouble differentiating this problem, it looks like the quotient rule is being used within the product rule, but I don't know how to combine the two of them.

    f(x)= ln(x+1/x-1) I started using the product rule but then I got lost when I had to differentiate the second term, because this would be a quotient rule in this term. If someone could illustrate how to differentiate this, Thanks...
    There are two ways:
    The direct approach:
    f^{\prime}(x) = \frac{1}{\frac{x + 1}{x - 1}} \cdot \frac{(1)(x - 1) - (x + 1)(1)}{(x - 1)^2}
    etc.

    or use the division property of logarithms:
    f(x) = ln(x + 1) - ln(x - 1)
    then take the derivative:
    f^{\prime}(x) = \frac{1}{x + 1} - \frac{1}{x - 1}

    The two methods have the same result. The second is obviously faster, but the first is better for algebra practice.

    -Dan
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Feb 2008
    Posts
    61
    I believe my professor wants us to use the first method you described, the answer in the book is: -2 / (x+1)(x-1) I don't see how this is obtained, I took the 1/(x+1)(x-1) times -2/(x-1)^2 so when I multiply the numerator matches the answer, however the denominator is different, I have ((x+1)/(x-1)) times (x-1)^2 and the book only has (x+1)(x-1) Are the (x-1) terms being cancelled when they are multiplied or what? Thanks again! Also what rules are you using? It doesnt look like the product rule, as there is no + sign between f g' and g f'.....So I am curious why in the first step you has 1/(x+1)/(x-1) isn't this term supposed to remain undifferentiated?
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Member
    Joined
    Mar 2008
    Posts
    148
    The derivative of log(x) is 1/x. He is using the chain rule.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Junior Member
    Joined
    Feb 2008
    Posts
    61
    Thank You for your help. I just would like to know if I have the right answer for the following problem:

    f(x) = square root of ln t+t

    I used the chain rule, got rid of the radical by raising the terms to the 1/2 power. I got as an answer: 1/2(ln t +t)^-1/2 ((1/t)+1) Is this correct?

    The book has answer as: t+1 / 2t(square root of ln t +t) Thanks....
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by kdogg121 View Post
    Thank You for your help. I just would like to know if I have the right answer for the following problem:

    f(x) = square root of ln t+t

    I used the chain rule, got rid of the radical by raising the terms to the 1/2 power. I got as an answer: 1/2(ln t +t)^-1/2 ((1/t)+1) Is this correct? .... Yes

    The book has answer as: t+1 / 2t(square root of ln t +t) Thanks....
    \frac12 (\ln(t)+t)^{-\frac12} \cdot \left(\frac1t + 1\right) = \frac{\frac1t + 1}{2 \cdot \sqrt{\ln(t)+t}} = \frac{\frac1t(1 + t)}{2 \cdot \sqrt{\ln(t)+t}} = \frac{t+1}{2t \cdot \sqrt{\ln(t)+t}}
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Exponential and Logarithmic Functions
    Posted in the Algebra Forum
    Replies: 2
    Last Post: December 12th 2010, 05:42 PM
  2. Exponential and Logarithmic Functions
    Posted in the Algebra Forum
    Replies: 1
    Last Post: November 13th 2010, 10:50 AM
  3. Exponential and Logarithmic Functions
    Posted in the Algebra Forum
    Replies: 3
    Last Post: February 23rd 2010, 03:04 PM
  4. exponential and logarithmic functions
    Posted in the Algebra Forum
    Replies: 3
    Last Post: July 29th 2008, 10:58 PM
  5. exponential and logarithmic functions
    Posted in the Algebra Forum
    Replies: 2
    Last Post: July 20th 2008, 03:04 PM

Search Tags


/mathhelpforum @mathhelpforum