# product rule and differentiate natural log function

• Apr 7th 2012, 01:42 PM
rabert1
product rule and differentiate natural log function
$\displaystyle g(x)= 3^x * ln(x)$

$\displaystyle g'(x)= (3^x)' (ln(x)) + (3^x) (ln(x))'$

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

$\displaystyle g'(x)= (3^xln3)(ln(x)) + \frac{3^x}{x}$ \\can be simplified or stop here?
• Apr 7th 2012, 01:54 PM
Plato
Re: product rule and differentiate natural log function
Quote:

Originally Posted by rabert1
$\displaystyle g(x)= 3^x * ln(x)$

$\displaystyle g'(x)= (3^x)' (ln(x)) + (3^x) (ln(x))'$

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

$\displaystyle g'(x)= (3^xln3)(ln(x)) + \frac{3^x}{x}$ \\can be simplified or stop here?

No. Just stop. But to be strictly correct it should be $\displaystyle \ln(3)$. It is a function.
• Apr 7th 2012, 09:19 PM
princeps
Re: product rule and differentiate natural log function
Alternative form :

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