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Math Help - product rule and differentiate natural log function

  1. #1
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    product rule and differentiate natural log function

     g(x)=  3^x * ln(x)

     g'(x)=  (3^x)' (ln(x)) + (3^x) (ln(x))'

     g'(x)=  (3^xln3)(ln(x)) + (3^x)(\frac{1}{x})

     g'(x)= (3^xln3)(ln(x)) + \frac{3^x}{x} \\can be simplified or stop here?
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  2. #2
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    Re: product rule and differentiate natural log function

    Quote Originally Posted by rabert1 View Post
     g(x)=  3^x * ln(x)

     g'(x)=  (3^x)' (ln(x)) + (3^x) (ln(x))'

     g'(x)=  (3^xln3)(ln(x)) + (3^x)(\frac{1}{x})

     g'(x)= (3^xln3)(ln(x)) + \frac{3^x}{x} \\can be simplified or stop here?
    No. Just stop. But to be strictly correct it should be \ln(3). It is a function.
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  3. #3
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    Re: product rule and differentiate natural log function

    Alternative form :

    g'(x)=3^x \cdot \ln x^{\ln 3}+3^x \cdot \ln e^{\frac{1}{x}}=3^x \cdot \ln \left(x^{\ln 3} \cdot e^{\frac{1}{x}}\right)
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