# logarithmic differentiation

• Feb 26th 2008, 08:59 AM
mortalapeman
logarithmic differentiation
Use logarithmic differentiation to find the derivative of the following equation.
$\displaystyle y = (ln(x))^(cos(15 x))$

in the home work i was doing it was a multiple choice question, and i ended up getting it right, but i'm not exactly sure how to get from the question to the answer

$\displaystyle y'= ln(x)^{cos(15x)}(cos(15x)/x(ln(x))-15(sin(15x))(ln(ln(x)))$

i worked out the problem and got everything in this answer except for the $\displaystyle (cos(15x)/x(ln(x))$ part, and i don't know where it came from in the derivative formulas that i have.

I'm asking in case there is a question like this on the test and i'll have no idea how to do it correctly.
• Feb 26th 2008, 09:55 AM
Peritus
$\displaystyle y = \ln x^{\cos 15x}$

here's a useful little trick:

$\displaystyle \ln y = \ln \left( {\ln x} \right)^{\cos 15x} = \cos 15x\ln \ln x$

now differentiate the equation:

$\displaystyle \begin{gathered} \frac{{y'}} {y} = \ln \ln x\frac{d} {{dx}}\cos 15x + \cos 15x\frac{d} {{dx}}\ln \ln x \hfill \\ \hfill \\ \Leftrightarrow y' = \ln v^{\cos 15x} \left( {\cos 15x\frac{1} {{x\ln x}} - 15\sin 15x\ln \ln x} \right) \hfill \\ \end{gathered}$