Question

Use logarithmic differentiation to find the derivative of the function.

Answer 1

\[y=(\cos(9x))^{x}\]

Answer 2

take the log get
\[x\ln(cos(9x))\]

Answer 3

then take the derivative using product and chain rule
\[\frac{d}{dx}x\ln(\cos(9x))=\ln(\cos(9x))-\frac{9\sin(9x)}{\cos(9x)}\]

Answer 4

then multiply the result by the original function to get your answer

Answer 5

oops i made a mistake. it should be
\[\ln(\cos(9x))-x\times \frac{9\sin(9x))}{\cos(9x)}\]

Answer 6

so " final answer" is
\[\cos(9x)^x(\ln(\cos(9x))-\frac{9x\sin(9x)}{\cos(9x)})\]

Answer 7

other equivalent way to do this is rewrite
\[\cos(9x)^x=e^{x\ln(cos(9x))}\] and then take the derivative using the chain and product rule, but all the work is the same as is the answer

Answer 8

last term could be rewritten as
\[-9x\tan(9x)\] if you like

Answer 9

they said its wrong

Answer 10

i am fairly certain of my answer.

Answer 11

i don't know who the "they" are but i checked my work and i don't see the usual typo. what did they say the answer was?

Answer 12

I got almost the same answer \[(\cos(9x))^{x} [\ln(\cos(9x))-9xtan(9x)]\]