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

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- Nov 4th 2008, 05:08 PMbigtonLOGS
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 - Nov 4th 2008, 06:01 PMbigton
Which rule would you use first

- Nov 4th 2008, 06:23 PMbigton
the problem is f(x) = 2^(log(3) x)

sorry about that - Nov 4th 2008, 06:39 PMakolman
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

$\displaystyle \frac{1}{y} y'=(\log{2})(\log {3})$

$\displaystyle y'=(\log{2})(\log {3})y=(\log{2})(\log {3})2^{(\log {3})x}$ - Nov 4th 2008, 06:53 PMSoroban
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)} $

Quote:

Find $\displaystyle f'(x)\!:\;\;f(x) \:=\:2^{\log_3(x)}$

$\displaystyle f'(x) \;=\;2^{\log_3(x)}\cdot\frac{1}{x\ln(3)}\ln(2) \;=\;\frac{2^{\log_3(x)}\ln(2)}{x\ln(3)}$