What is the derivative of `cos^(-1)((2cosx+3sinx)/(sqrt(13)))`?

Options

A. `(1)/(sqrt(1-x^(2)))`
B. `-(1)/(sqrt(1-x^(2)))`
C. 0
D. 1

Answer

D

Explanation

The given function `f(x)=cos^(-1)((2cosx+3sinx)/(sqrt(13))}`
can be written as
`cos^(-1)(cosx.(2)/(sqrt(13))+sinx.(3)/(sqrt(13))}" ...(i)"`
Let `(2)/(sqrt(13))=cos theta and (3)/(sqrt(13))=sin theta`
`rArrtantheta=((3)/(sqrt(13)))/((2)/(sqrt(13)))=(3)/(2)`
So, (i), `cos^(-1)(cos x cos theta+sin x sin theta)`
where `theta=tan^(-1)((3)/(2))`
`=cos^(-1)(cos(x-theta))=x-theta`
hence, `f'(x)=1" "(because theta " is a constant")`