Let `y(x)=cos(3cos^-1x),x in [-1,1],x != +- sqrt3/2.` Then `1/(y(x)){(x^2-1)(d^2y(x))/(dx^2)+x(dy(x))/(dx)}` equals

Options

A. 1
B. 2
C. 8
D. 9

Answer

D

Explanation

We have, `y=cos(3cos^(-1)x)`
`implies" "(dy)/(dx)=(3sin(3cos^(-1)x))/(sqrt(1-x^(2)))`
`implies" "sqrt(1-x^(2))(dy)/(dx)=3sin(3cos^(-1)x)`
`implies" "(-x)/(sqrt(1-x^(2)))(dy)/(dx)+sqrt(1-x^(2))(d^(2)y)/(dx^(2))=3cos(3cos^(-1)x)xx-(3)/(sqrt(1-x^(2)))`
`implies" "-x(dy)/(dx)+(1-x^(2))(d^(2)y)/(dx^(2))=-9y`
`implies" "(1)/(y){(x^(2)-1)(d^(2)y)/(dx^(2))+x(dy)/(dx)}=9`
ALITER We have,
`y=cos(3cos^(-1)x)-cos(cos^(-1)(4x^(3)-3x))=4x^(3)-3x`
`implies" "(dy)/(dx)=12x^(2)-3" and "(d^(2)y)/(dx^(2))=24x`
`:." "(1)/(y){(x^(2)-1)(d^(2)y)/(dx^(2))+x(dy)/(dx)}=(1)/(y){(x^(2)-1)(24x)+x(12x^(2)-3)}`
`=(1)/(y)(36x^(3)-27x)=(9y)/(y)=9`