Question

Find the derivative of the function?

`y=ln tan^2 3x`

Answer 1

`dy/dx=12csc(6x)`

Explanation:

We to find the derivative of

`y=ln(tan^2(3x))=2ln(tan(3x))`

Use the chain rule, if `y=f(u)` and `u=g(x)` then

`dy/dx=dy/(du)(du)/dx`

Let `y=2ln(u)=>(dy)/(du)=2/u`

and `u=tan(3x)=>(du)/(dx)=3sec^2(3x)`

Thus

`dy/dx=6/usec^2(3x)=6/tan(3x)sec^2(3x)=6csc(3x)sec(3x)`

Or by applying the identity `csc(2x)=1/2csc(x)sec(x)`

`dy/dx=12csc(6x)`