d/dx ((ln(x) + log 2 (x))/(tan(x))

f(x)  =  [ lnx + log₂ x ] / [tanx]    =

lnx / tanx    +  log₂ x / tanx      using the Quotient Rule  on both, we have

[ (1/x) * tanx -  lnx * sec^2 x]  / [tan^2 x]     + [ 1 / [x ln 2] * tanx - log₂ x * sec^2x] / [tan^2x]  =

[ tanx/x - lnx * [tan^2 x + 1] / [tan^2 x]  + ( [ tanx/ [ x ln2] -  log₂ x [ 1 + tan^2x] ) / [tan^2 x]  =

[ tanx ( 1/x + 1/[ x ln2] )  - [ tan^2x + 1] ( lnx + log₂ x ) ] /  [ tan^2 x]  =

[ ln 2 + 1] / [ x* ln2 * tanx]  -  [ sec^2x / tan^2x] * (lnx + log₂ x)  =

[ ln 2 + 1] / [ x* ln2 * tanx]   -  csc^2x * (ln x + log₂ x )  =

[ln2 + 1]  * cot x  / [x ln 2]  -  csc^2x (ln x + log₂ x)