Results 1 to 10 of 10

Math Help - Differentiate and Simplify

  1. #1
    Newbie
    Joined
    Jan 2012
    Posts
    3

    Differentiate and Simplify




    Differentiate the following functions and simplify:

    a) (x^4+2x^1/2)(lnx)

    b) x^(x^2)

    c) sinx/x^6

    d) (e^4x^2)-2.
    N:B The parentheses is not in the question, I only used it to separate -2 from x as it is not part of the power of x and to make the question clearer

    2) Determine the derivative of the inverse f^-1 of f(x) =3^(x/2)+logx(base 2)+(x^3) -7 at the point 5=f(2).
    Write down the equation of the tangent line of f^-1 at 5.

    Thanks in advance.
    Happy New Year!!!!!!!!!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Amer's Avatar
    Joined
    May 2009
    From
    Jordan
    Posts
    1,093

    re: Differentiate and Simplify

    you should know the chain rule and how to derive a fraction of functions or multiples of functions
    y = f(x)g(x)

    y'= f'(x)g(x) + g'(x)f(x)
    if
    y = \frac{f(x)}{g(x)}

    y' = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}

    the second one you need to make a trick let

     u = x^{x^2}
    take the ln
    \ln u = \ln x^{x^2}

    \ln u = x^2 \ln x derive both sides

    show some work
    Follow Math Help Forum on Facebook and Google+

  3. #3
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    re: Differentiate and Simplify

    2) You have to use:
    \frac{df^{-1}}{dx}(f(x))=\frac{1}{f'(x)}
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Newbie
    Joined
    Jan 2012
    Posts
    3

    re: Differentiate and Simplify

    @Siron, thanks for your reply but i still can't solve them.
    I need you to put me through.
    in number 2, the log is to base 2.
    how do i even start to solve that?
    Thanks
    Follow Math Help Forum on Facebook and Google+

  5. #5
    -1
    e^(i*pi)'s Avatar
    Joined
    Feb 2009
    From
    West Midlands, England
    Posts
    3,053
    Thanks
    1

    re: Differentiate and Simplify

    Quote Originally Posted by Tizi View Post
    @Siron, thanks for your reply but i still can't solve them.
    I need you to put me through.
    in number 2, the log is to base 2.
    how do i even start to solve that?
    Thanks
    Use the change of base rule: \log_2(x) = \dfrac{\ln(x)}{\ln(2)}
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Jan 2012
    Posts
    3

    re: Differentiate and Simplify

    in a) i already solved with chain rule
    here is what i did.
    (x^4+2x^1/2)(lnx)
    4x^3+x^(-1/2)*lnx + (x^4+2x^1/2)*1/x
    how can i simplify this?

    b) here's what i did as you have said
    y=x^(x^2)
    lny=ln x^(x^2)
    1/y=x^2lnx
    using chain rule,
    1/y= x^2*1/x+2x*lnx
    1/y=x+2xlnx
    1/y=x(1+2lnx)
    it doesn't seem right to me, i think i have missed something.
    i need help!
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Re: Differentiate and Simplify

    Quote Originally Posted by Tizi View Post
    @Siron, thanks for your reply but i still can't solve them.
    I need you to put me through.
    in number 2, the log is to base 2.
    how do i even start to solve that?
    Thanks
    You want to find the derivative of f^{-1} at the point f(2)=5 so you have to find:
    \frac{df^{-1}}{dx}(5)=\frac{1}{f'(2)}

    That means you have to calculate f'(x) ...
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor Amer's Avatar
    Joined
    May 2009
    From
    Jordan
    Posts
    1,093

    Re: Differentiate and Simplify

    Quote Originally Posted by Tizi View Post
    in a) i already solved with chain rule
    here is what i did.
    (x^4+2x^1/2)(lnx)
    4x^3+x^(-1/2)*lnx + (x^4+2x^1/2)*1/x
    how can i simplify this?

    b) here's what i did as you have said
    y=x^(x^2)
    lny=ln x^(x^2)
    1/y=x^2lnx
    using chain rule,
    1/y= x^2*1/x+2x*lnx
    1/y=x+2xlnx
    1/y=x(1+2lnx)
    it doesn't seem right to me, i think i have missed something.
    i need help!
    for the first one the second term \frac{x^4+2 x^{\frac{1}{2}}}{x} =x^3 + 2 x^{\frac{-1}{2}}

    for the second one you have to becareful
    \ln y = x^2 \ln x

    \frac{y'}{y} = x + 2x \ln x
    y = x^2 ln x

     y' = y ( x + 2x \ln x ) = (x^2 \ln x )(x + 2x \ln x )
    Follow Math Help Forum on Facebook and Google+

  9. #9
    MHF Contributor Amer's Avatar
    Joined
    May 2009
    From
    Jordan
    Posts
    1,093

    Re: Differentiate and Simplify

    Quote Originally Posted by Siron View Post
    2) You have to use:
    \frac{df^{-1}}{dx}(f(x))=\frac{1}{f'(x)}
    I would like to see a proof for it, Thanks
    Follow Math Help Forum on Facebook and Google+

  10. #10
    MHF Contributor Siron's Avatar
    Joined
    Jul 2011
    From
    Norway
    Posts
    1,250
    Thanks
    20

    Re: Differentiate and Simplify

    Quote Originally Posted by Amer View Post
    I would like to see a proof for it, Thanks
    Take a look here:
    Inverse function theorem - Wikipedia, the free encyclopedia
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Simplify then differentiate times 3
    Posted in the Calculus Forum
    Replies: 1
    Last Post: May 5th 2010, 10:26 AM
  2. differentiate and simplify #1
    Posted in the Calculus Forum
    Replies: 6
    Last Post: August 9th 2009, 05:14 PM
  3. differentiate and simplify #2
    Posted in the Calculus Forum
    Replies: 9
    Last Post: August 9th 2009, 03:08 PM
  4. differentiate and simplify #5
    Posted in the Calculus Forum
    Replies: 4
    Last Post: August 7th 2009, 07:31 AM
  5. Differentiate, Do not simplify
    Posted in the Calculus Forum
    Replies: 0
    Last Post: November 16th 2008, 09:18 AM

Search Tags


/mathhelpforum @mathhelpforum