Question

What is the derivative of `f(x)=ln(tan(x))` ?

Answer 1

`f'(x)=2(cosec2x)`

Solution

`f(x)=ln(tan(x))`

let's begin with general example, suppose we have

`y=f(g(x))`

then, Using Chain Rule,

`y'=f'(g(x))*g'(x)`

Similarly following the given problem,

`f'(x)=1/tanx*sec^2x`

`f'(x)=cosx/sinx*1/(cos^2x)`

`f'(x)=1/(sinxcosx)`

for simplifying further, we multiply and divide by 2,

`f'(x)=2/(2sinxcosx)`

`f'(x)=2/(sin2x)`

`f'(x)=2(cosec2x)`