Given y=log{tanx}
from the definition of logarithm we get
tanx = 10^y
by appliying log(base)e or {ln}
ln{tanx} = y ln10
differentiating both sides
sec^2(x)/tanx = ln10 dy/dx
1/{tanxsec^2x} = ln10 dy/dx
1{sinxcosx} = ln10 dy/dx
= 2/{2sinxcosx} = ln10 dy/dx --------------(multiplying 2 on the denominator and numerator of the L.H.S)
=2/sin(2x) = ln10 dy/dx ..................... = 2cosec(2x) =ln10 dy/dx
dy/dx ={2cosec(2x)}/ln10