Question

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

Answer 1

`f'(x) = -sin(ln(sinx))(1/sinx cosx) = cotxsin(ln(cscx))`

Explanation:

Use the chain rule:

`f'(x) = -sin(ln(sinx))[d/dx(ln(sinx))]`

`= -sin(ln(sinx))[1/sinx d/dx(sinx)]`

`= -sin(ln(sinx))[1/sinx cosx]`

`= sin(-ln(sinx)) cotx`

`= cotxsin(ln(1/sinx))`

`= cotxsin(ln(cscx))`