Differentiate the functions with respect to `x :` `cos^(-1){(cosx+sinx)/(sqrt(2))},-pi/4

Options

A. 0
B. 1
C. -1
D. 2

Answer

C

Explanation

Let `y= cos^(-1) {(cos x + sin x)/(sqrt(2))}`
`y = cos^(-1) {cosxx((1)/(sqrt(2))) + sinxx((1)/(sqrt(2))) }`
`cos xx cos ((pi)/(4)) + sin xx((pi)/(4))}`
`y = cos^(-1) [cos(x -(pi)/(4))]` ...(i)
Here, `-(pi)/(4) lt xlt(pi)/4`
`rArr (-(pi)/(4)-(pi)/(4)) lt (x - (pi)/(4)) lt ((pi)/(4)- (pi)/(4))`
`rArr -(pi)/(2)lt (x- (pi)/(4)) lt 0 `
So, from Eq. (i) , ` y = - (x - (pi)/(4))`
` [ because^(-1) (cos theta ) = - theta, if theta in [ - pi, 0]]`
On differentiating both sieds w.r.t.x., we get `(dy)/(dx) = - 1`