Use logarithmic differentiation with all properties of logarithms

Question

If r(x)=(x⁶cosx)/sinx, find r′(x) using logarithmic differentiation.

Answer 1

y = x⁶cosx/sinx

lny = 6lnx + ln(cosx) - ln(sinx)

y'/y = 6/x - sinx/cosx - cosx/sinx

y' = x⁶cosx/sinx(6/x - sinx/cosx - cosx/sinx)

y' = 6x⁵cotx - x⁶ - x⁶cot²x