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Math Help - Help with derivatives of these functions

  1. #1
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    Help with derivatives of these functions

    1. f(x)= e^(2*pi*x)
    2. f(x)= x^(2*pi*e)
    3. f(x)= (e*pi)^(2x)
    4. f(x)= pi^(e)^(2x)

    thanks
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  2. #2
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    Quote Originally Posted by Dr. Noobles View Post
    1. f(x)= e^(2*pi*x)
    2. f(x)= x^(2*pi*e)
    Recall that \frac{d}{dx}e^{x} = e^{x}.

    The first is going to be done by the chain rule:
    1. f(x)= e^{2 \pi x}

    f'(x) = e^{2 \pi x} \cdot 2 \pi = 2 \pi e^{2 \pi x}

    Since \pi and "e" are just constants, the second one is familiar to you already. It is just the power law:
    \frac{d}{dx}x^n = nx^{n-1}

    2. f(x)= x^{2 \pi e}

    f'(x) = (2 \pi e) x^{2 \pi e - 1}

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Dr. Noobles View Post
    3. f(x)= (e*pi)^(2x)
    Since \pi and "e" are constants this is the derivative of a constant to a function:
    \frac{d}{dx}a^x = ln(a) \cdot a^x

    (We also need to use the chain rule.)

    f(x) = (e \pi)^{2x}

    f'(x) = \left ( ln(e \pi) \cdot (e \pi)^{2x} \right ) \cdot 2

    f'(x) = 2 \left ( ln(e) + ln( \pi ) \right ) (e \pi )^{2x} Using a property of logarithms.

    f'(x) = 2 \left ( 1 + ln( \pi ) \right ) (e \pi )^{2x}

    -Dan
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Dr. Noobles View Post
    4. f(x)= pi^(e)^(2x)
    We need some clarification here. Is your function:
    f(x) = \pi ^{ \left ( e^{2x} \right ) }

    or

    f(x) = \left ( \pi ^{e} \right )^{2x}

    For the first case, we need to use the chain rule a few times:
    f(x) = \pi ^{ \left ( e^{2x} \right ) }

    f'(x) = \left ( ln( \pi ) \cdot \pi ^{ \left ( e^{2x} \right ) } \right ) \cdot \left ( e^{2x} \right ) \cdot (2)

    f'(x) = 2 ln( \pi ) e^{2x} \cdot \pi ^{ \left ( e^{2x} \right ) }

    In the second case note that \left ( \pi ^{e} \right )^{2x} = \pi ^{2ex}, so
    f(x) = \left ( \pi ^{e} \right )^{2x} = \pi ^{2ex}

    f'(x) = \left ( ln( \pi ) \cdot \pi ^{2ex} \right ) \cdot (2e)

    f'(x) = 2e \cdot ln( \pi ) \cdot \pi ^{2ex}

    -Dan
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