Question

find dy/dx if y= (cos x)^x

Answer 1

hmm

Answer 2

method one
take the log
use property of log to simplify
take the derivative using the product rule and the chain rule
multiply by the original function

method two
write \(\cos(x)^x\) as \(e^{x\ln(\cos(x))}\) and take the derivative using the chain rule
the work is identical

Answer 3

I guess I've seen this question today..
y= (cos x)^x
Take ln on both sides
ln y = x ln cosx
Diff both sides w.r.t. x
(1/y) dy/dx = x (1/cosx) (-sinx) + lncosx
dy/dx = y(- xtanx + lncosx)
= cosx^x (lncosx-xtanx )
It should be something like this :S