Results 1 to 3 of 3

Math Help - Derivative of a fractional exponent proof?

  1. #1
    Junior Member
    Joined
    Apr 2010
    Posts
    26

    Derivative of a fractional exponent proof?

    such that if
    f(x) = x^{p/q} then f'(x)={p/q}x^{p/q-1}

    using the definition of a derivative (the 'lim as h tends to zero' one), the formula a^{n}-b^{n}=(a-b)(a^{n-1}+a^{n-1}b+...+b^{n-1})
    and fact that the derivative of f(x)=x^{1/q} is 1/qx^{1/q-1} (proved earlier using the above formula, where the "h" falls out nicely)

    I have achieved the result implicitly, but did not use the def-n of the derivative or the a^{n}-b^{n} formula
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Prove It's Avatar
    Joined
    Aug 2008
    Posts
    10,969
    Thanks
    1009
    You can prove that \displaystyle \frac{d}{dx}(x^n) = n\,x^{n-1} for all \displaystyle n using the chain rule.


    \displaystyle y = x^n

    \displaystyle = e^{\ln{(x^n)}}

    \displaystyle = e^{n\ln{x}}.


    Now let \displaystyle u = n\ln{x} so that \displaystyle y = e^u.


    \displaystyle \frac{du}{dx} = \frac{n}{x}.


    \displaystyle \frac{dy}{du} = e^u = e^{n\ln{x}}.


    Thus \displaystyle \frac{dy}{dx} = \frac{du}{dx}\,\frac{dy}{du}

    \displaystyle = \frac{n\,e^{n\ln{x}}}{x}

    \displaystyle = \frac{n\,e^{\ln{(x^n)}}}{x}

    \displaystyle = \frac{n\,x^n}{x}

    \displaystyle = n\,x^{n-1}.

    Q.E.D.


    Of course, you will need to have proved the chain rule and the derivatives of \displaystyle e^x and \displaystyle \ln{x} beforehand...
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Apr 2010
    Posts
    26
    thank you for this, but i am required to construct a proof from first principles and only using the information given
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 2
    Last Post: December 19th 2010, 08:04 AM
  2. fractional exponent simplification
    Posted in the Algebra Forum
    Replies: 3
    Last Post: April 1st 2010, 01:31 PM
  3. fractional exponent derivative
    Posted in the Calculus Forum
    Replies: 3
    Last Post: December 14th 2009, 02:09 PM
  4. Simplifying a fractional base to exponent
    Posted in the Algebra Forum
    Replies: 3
    Last Post: October 27th 2009, 08:52 PM
  5. Req. Help with Fractional Exponent Problem
    Posted in the Algebra Forum
    Replies: 2
    Last Post: January 24th 2008, 04:30 AM

Search Tags


/mathhelpforum @mathhelpforum