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Proof of derivative a^x

  1. Apr 13, 2012 #1
    [Answered] Proof of derivative a^x

    [Answered] Thanks.

    I get a wrong expression when I try to take to take the derivative of (ln a) x

    1. The problem statement, all variables and given/known data
    Prove [tex]\frac{d}{dx} a^x = (ln a)a^x[/tex]


    2. Relevant equations
    By definition, [tex]a^x = e^{e(ln a)x}[/tex]


    3. The attempt at a solution
    [tex]\frac{d}{dx} a^x = (ln a)a^x[/tex]
    [tex]\frac{d}{dx}e^u\frac{du}{dx}[/tex]
    Solving for du/dx gives [tex]\frac{d}{dx}(ln a)x[/tex]
    Let [tex] u = ln a . v = x[/tex]
    From the product rule,
    [tex]u'v+v'u[/tex] which gives
    [tex]\frac{du}{dx} = \frac{x}{a} + ln a [/tex]
    so plugging it back to original expression gives
    [tex]\frac{d}{dx}a^x = e^{e(ln a)x} * (\frac{x}{a} + ln a)[/tex]

    Which is obviously wrong.
     
    Last edited: Apr 13, 2012
  2. jcsd
  3. Apr 13, 2012 #2
    This should be [tex]a^x = (e^{ln(a)})^{x} = e^{(ln(a)) x}. [/tex] Now taking the derivate with respect to x should bring you directly to your answer (remember that ln(a) is a constant).
     
  4. Apr 13, 2012 #3
    Whoops, a is a constant.. it completely skipped my mind. Thank you!!
     
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