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Math Help - Differentiation

  1. #1
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    Differentiation

    Hi I dont have an actual question but if I wanted to how can i differentiate a function like y=3^(x) with respect to x?

    thanks
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    Quote Originally Posted by wahhdoe View Post
    Hi I dont have an actual question but if I wanted to how can i differentiate a function like y=3^(x) with respect to x?

    thanks
    This is of the form a^x

    \frac{d}{dx}a^x=a^xlog_ea

    \frac{d}{dx}3^x=3^xln3
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    Quote Originally Posted by wahhdoe View Post
    Hi I dont have an actual question but if I wanted to how can i differentiate a function like y=3^(x) with respect to x?

    thanks
    If you don't want to memorize the formulas just take the log's.

    y=3^x

    becomes \ln y=x\ln 3.

    The ln 3 is a constant, so when you differentiate with respect to x you get, using the chain rule

    {1\over y}{dy\over dx}=\ln 3

    Popping the y on the other sides gives you {dy\over dx}=y\ln 3=3^x\ln 3.
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  4. #4
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    Quote Originally Posted by wahhdoe View Post
    Hi I dont have an actual question but if I wanted to how can i differentiate a function like y=3^(x) with respect to x?

    thanks
    Alternatively, transform it into a function of e.


    y = 3^x

     = e^{\ln{3^x}}

     = e^{x\ln{3}}

     = (e^x)^{\ln{3}}.


    Let u = e^x so that y = u^{\ln{3}}.


    \frac{du}{dx} = e^x


    \frac{dy}{du} = (\ln{3})u^{\ln{3} - 1}

     = (\ln{3})(e^x)^{\ln{3} - 1}

     = (\ln{3})e^{x\ln{3} - x}

     = (\ln{3})(e^{x\ln{3}})(e^{-x})

     = (\ln{3})(e^{\ln{3^x}})(e^{-x})

     = (\ln{3})(3^x)(e^{-x}).


    Therefore

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

     = e^x(\ln{3})(3^x)(e^{-x})

     = (\ln{3})3^x.
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