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Math Help - Differentiation

  1. #1
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    Differentiation

    Differentiate each of the following with respect to x. Simplify fully when indicated.

    1)
    y = e^{3x - 2} + ln(x^2 + 2x)

    Attempt:
    y' = 3e^{3x - 2} + ???

    I don't know how to differentiate for ln(x^2 + 2x)

    Answer:
    There is no answer in the book...

    ----------


    2) y = 3^{3x} - log_{10}(3x - 1)

    Attempt:
    y' = 3^{3x}ln3 - ???

    I don't know how to differentiate for - log_{10}(3x - 1)

    Answer:
    y' = 3e^{3x - 2} + 3^{3x + 1}ln3


    ----------


    3) y = \frac {e^{x}}{e^{x} - e^{-x}}

    Attempt:
    y' = \frac {e^{x}(e^{x} - e^{-x}) - (e^{x} - e^{-x}(-2))(e^{x})}{(e^{x} - e^{-x})^2}
    y' = \frac {e^{x}(e^{x} - e^{-x}) - e^{x}(e^{x} + 2e^{-x})}{(e^{x} - e^{-x})^2}

    I'm not sure how to simplify it

    Answer:
    y' = \frac{-2}{(e^{x} - e^{-x})^2}


    ----------


    4) y = sin^{3}(3x - 1) - cos^2(x^2 - 1)

    Attempt:
    I tried to use the product rule...

    y' = 3cos^2(3x - 1) + 3sin^{3} - 2sin(x^2 - 1) - 2cos^2
    y' = 9cos^{2}x - 3cos^2x + 3sin^{3} - 2sin^2x + 2sin - 2cos^2x

    Answer:
    y' = 9sin^2(3x - 1)cos(3x - 1) + 4xcos(x^2 - 1)(sin(x^2 - 1)
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by Macleef View Post
    Differentiate each of the following with respect to x. Simplify fully when indicated.

    1)
    y = e^{3x - 2} + ln(x^2 + 2x)

    Attempt:
    y' = 3e^{3x - 2} + ???

    I don't know how to differentiate for ln(x^2 + 2x)

    Answer:
    There is no answer in the book...

    ----------


    2) y = 3^{3x} - log_{10}(3x - 1)

    Attempt:
    y' = 3^{3x}ln3 - ???

    I don't know how to differentiate for - log_{10}(3x - 1)

    Answer:
    y' = 3e^{3x - 2} + 3^{3x + 1}ln3


    ----------


    3) y = \frac {e^{x}}{e^{x} - e^{-x}}

    Attempt:
    y' = \frac {e^{x}(e^{x} - e^{-x}) - (e^{x} - e^{-x}(-2))(e^{x})}{(e^{x} - e^{-x})^2}
    y' = \frac {e^{x}(e^{x} - e^{-x}) - e^{x}(e^{x} + 2e^{-x})}{(e^{x} - e^{-x})^2}

    I'm not sure how to simplify it

    Answer:
    y' = \frac{-2}{(e^{x} - e^{-x})^2}


    ----------


    4) y = sin^{3}(3x - 1) - cos^2(x^2 - 1)

    Attempt:
    I tried to use the product rule...

    y' = 3cos^2(3x - 1) + 3sin^{3} - 2sin(x^2 - 1) - 2cos^2
    y' = 9cos^{2}x - 3cos^2x + 3sin^{3} - 2sin^2x + 2sin - 2cos^2x

    Answer:
    y' = 9sin^2(3x - 1)cos(3x - 1) + 4xcos(x^2 - 1)(sin(x^2 - 1)
    \frac{d}{dx}\bigg[\ln(u(x))\bigg]=\frac{u'(x)}{u(x)}

    So \frac{d}{dx}\bigg[\log_a(u(x))\bigg]=\frac{1}{\ln(a)}\frac{d}{dx}\bigg[\ln(u(x))\bigg]=\frac{u'(x)}{\ln(a)u(x)}
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  3. #3
    Moo
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    Hello,

    In general, the derivative of \ln(ax+b) is \frac{a}{ax+b} (this is chain rule)

    1) For \ln(x^2+2x), note that this is \ln[x(x+2)]=\ln x+\ln(x+2)

    Then, differentiate


    2) \log_a(x)=\frac{\ln(x)}{\ln(a)}, where \ln(a) is obviously a constant


    3) Just expand !
    e^xe^x=e^{2x}, using the rule for exponents.


    4) Let f(x)=\sin(3x-1). f'(x)=3 \cos(3x-1)

    You want to differentiate (f(x))^3.

    The chain rule would yield :

    3 \cdot f'(x) \cdot (f(x))^2


    etc... ^^
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