differentiate 2^x

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  1. uxa595's Avatar
    1 09-05-2012  15:24
    • Peer Of The TSR Realm
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    differentiate 2^x
    how would you differentiate 2^x
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  2. Coulson72's Avatar
    2 09-05-2012  15:25
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    Re: differentiate 2^x
    2^x = e^xln2
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  3. raheem94's Avatar
    3 09-05-2012  15:25
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    Re: differentiate 2^x
    (Original post by uxa595)
    how would you differentiate 2^x
    Remember differentiating, y =a^x , gives, \displaystyle \frac{dy}{dx} = a^x lna
  4. uxa595's Avatar
    4 09-05-2012  15:28
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    Re: differentiate 2^x
    (Original post by Coulson72)
    2^x = e^xln2
    I got this but i don't think it's right

    (Original post by raheem94)
    Remember differentiating, y =a^x , gives, \displaystyle \frac{dy}{dx} = a^x lna
    How do you know a^x foes to a^x lna?
  5. f1mad's Avatar
    5 09-05-2012  15:30
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    Re: differentiate 2^x
    (Original post by nm786)
    remeber: dy/dx=x^n=nx^n-1
    2^x=(1)2x^(1-1)=2*1=2
    No.
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  6. raheem94's Avatar
    6 09-05-2012  15:32
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    Re: differentiate 2^x
    (Original post by uxa595)
    I got this but i don't think it's right



    How do you know a^x foes to a^x lna?
    Its a formula.

    If y= a^x
    Take the natural log of both sides.
    lny = lna^x \implies lny=xlna
    Differentiate implicitly,
    \displaystyle \frac1y \frac{dy}{dx} = lna \implies \frac{dy}{dx} = ylna

    We know, y=a^x , hence, \displaystyle \frac{dy}{dx} = ylna \implies \frac{dy}{dx} = a^x lna

    Do you get it?
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  7. f1mad's Avatar
    7 09-05-2012  15:33
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    Re: differentiate 2^x
    (Original post by uxa595)
    I got this but i don't think it's right



    How do you know a^x foes to a^x lna?
    If y= a^x

    ln(y)= xln(a)

    1/y(dy/dx)= ln(a)

    dy/dx= a^x(ln(a))
  8. raheem94's Avatar
    8 09-05-2012  15:34
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    Re: differentiate 2^x
    (Original post by nm786)
    is this for c1?
    no
  9. raheem94's Avatar
    9 09-05-2012  15:34
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    Re: differentiate 2^x
    (Original post by f1mad)
    If y= a^x

    ln(y)= xln(a)

    1/y(dy/dx)= ln(a)

    dy/dx= a^x(ln(a))
    My is better
  10. uxa595's Avatar
    10 09-05-2012  15:36
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    Re: differentiate 2^x
    I got this:

    y=2^x
    therefore
    y= e^(xln2)
    y= e^x . e^ln2
    y= 2e^x

    is that right?
  11. siwelmail's Avatar
    11 09-05-2012  15:38
    • Overlord in Training
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    Re: differentiate 2^x
    (Original post by nm786)
    is this for c1?
    C1? Holy **** I haven't done this for C4.
  12. raheem94's Avatar
    12 09-05-2012  15:41
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    Re: differentiate 2^x
    (Original post by uxa595)
    I got this:

    y=2^x
    therefore
    y= e^(xln2)
    y= e^x . e^ln2
    y= 2e^x

    is that right?
    No,

    Example:

    \displaystyle y=3^x = e^{ln3^x} = e^{xln3} \\ \frac{dy}{dx} = ln3 \times e^{xln3} = ln3 \times e^{ln3^x} = 3^xln3

    This could have directly been found using the formula i gave you.
  13. notnek's Avatar
    13 09-05-2012  15:41
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    Re: differentiate 2^x
    (Original post by raheem94)
    Its a formula.

    If y= a^x
    Take the natural log of both sides.
    lny = lna^x \implies lny=xlna
    Differentiate implicitly,
    \displaystyle \frac1y \frac{dy}{dx} = lna \implies \frac{dy}{dx} = ylna

    We know, y=a^x , hence, \displaystyle \frac{dy}{dx} = ylna \implies \frac{dy}{dx} = a^x lna

    Do you get it?
    Or... (if you're not fond of implicit differentiation):

    \displaystyle y=a^x \implies y=e^{\ln a^x} = e^{x\ln a}

    \displaystyle \implies \frac{dy}{dx} = \ln a \ e^{x\ln a} = a^x \ln a
  14. f1mad's Avatar
    14 09-05-2012  15:41
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    Re: differentiate 2^x
    (Original post by uxa595)
    I got this:

    y=2^x
    therefore
    y= e^(xln2)
    y= e^x . e^ln2
    y= 2e^x

    is that right?
    No.

    If y= e^(xln2)

    dy/dx= ln2*e^(xln2)

    And you should remember: e^(xln2)= 2^x.
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  15. raheem94's Avatar
    15 09-05-2012  15:42
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    Re: differentiate 2^x
    (Original post by notnek)
    Or... (if you're not fond of implicit differentiation):

    \displaystyle y=a^x \implies y=e^{\ln a^x} = e^{x\ln a}

    \displaystyle \implies \frac{dy}{dx} = \ln a \ e^{x\ln a} = a^x \ln a
    See post#14 in this thread, i know it mate.
  16. uxa595's Avatar
    16 09-05-2012  15:43
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    Re: differentiate 2^x
    (Original post by raheem94)
    No,

    Example:

    \displaystyle y=3^x = e^{ln3^x} = e^{xln3} \\ \frac{dy}{dx} = ln3 \times e^{xln3} = ln3 \times e^{ln3^x} = 3^xln3

    This could have directly been found using the formula i gave you.
    Ahhh right, thanks, i've got it now

    I can't remember doing this ever in class
  17. notnek's Avatar
    17 09-05-2012  15:47
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    • Location: Bangkok, Thailand
    Re: differentiate 2^x
    (Original post by raheem94)
    See post#14 in this thread, i know it mate.
    I assumed you knew it. I posted it for the OP.
  18. kingkongjaffa's Avatar
    18 09-05-2012  18:21
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    • Location: Manchester
    • Posts: 930
    Re: differentiate 2^x
    Just tried this myself

    Doing it implicitly seems the most straight forward
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