Question

find dy/dx if y=cos(x)

Answer 1

y= (cos x) ^ x
it is easier if there is no index
so take ln

ln y = ln (cosx)^x
lny=xlncosx
diff. both sides w.r.t. x
(1/y)(dy/dx) = lncosx+ x(1/cosx)(-sinx)
=ln cosx - tanx
dy/dx=(ln cosx - tanx)(y)
=(ln cosx - tanx)( (cos x) ^ x)
amendment:
(1/y)(dy/dx) = lncosx+ x(1/cosx)(-sinx)
=ln cosx - xtanx
dy/dx=(ln cosx -x tanx)(y)
=(ln cosx - xtanx)( (cos x) ^ x)