Find f'(x)
e) f(x) 2^(log(3)x)
The 3 is subscript
Guys this is the last problem for my homework help me through this
So you have something like this
$\displaystyle y=2^{(\log {3})x}$ and you want to find $\displaystyle y'$
$\displaystyle \log {y}=\log {(2^{(\log {3})x})}=(\log{2})(\log {3})x$
Now differentiate both sides
$\displaystyle \frac{1}{y} y'=(\log{2})(\log {3})$
$\displaystyle y'=(\log{2})(\log {3})y=(\log{2})(\log {3})2^{(\log {3})x}$
Hello, bigton!
We're expected to know these two formulas:
. . $\displaystyle f(x) \:=\:b^u \quad\Rightarrow\quad f'(x) \:=\:b^u\,u'\ln(b) $
. . $\displaystyle f(x) \:=\:\log_b(u) \quad\Rightarrow\quad f'(x) \:=\:\frac{1}{u\ln(b)} $
Find $\displaystyle f'(x)\!:\;\;f(x) \:=\:2^{\log_3(x)}$
$\displaystyle f'(x) \;=\;2^{\log_3(x)}\cdot\frac{1}{x\ln(3)}\ln(2) \;=\;\frac{2^{\log_3(x)}\ln(2)}{x\ln(3)}$