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If y=(t a n^(-1)\ x^2) , show that (x^2+...

If `y=(t a n^(-1)\ x^2)` , show that `(x^2+1)^2(d^2\ y)/(dx^2)+2x(x^2+1)(dy)/(dx)=2.`

Text Solution

Verified by Experts

`y=(tan^(-1)x)^(2)" ... (1)"`
Differentiating w.r.t. x, we get,
`(dy)/(dx)=(d)/(dx)(tan^(-1)x)^(2)=2tan^(-1)x.(d)/(dx)(tan^(-1)x)=2tan^(-1)x xx(1)/(1+x^(2))`
`therefore(1+x^(2))(dy)/(dx)=2tan^(-1)x`
`therefore(1+x^(2))^(2)((dy)/(dx))^(2)=4(tan^(-1)x)^(2)`
`therefore(1+x^(2))^(2)((dy)/(dx))^(2)=4(tan^(-1)x)^(2)`
`therefore(1+x^(2))^(2)((dy)/(dx))^(2)=4y" ... [By (1)]"`
Differentiating again w.r.t. x, we get,
`(1+x^(2))^(2).(d)/(dx)((dy)/(dx))^(2)+((dy)/(dx))^(2).(d)/(dx)(1+x^(2))^(2)=4(dy)/(dx)`
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Knowledge Check

  • IF y= ( tan ^(-1) x)^(2) " then " (x^(2) +1)^(2) (d^(2)y)/(dx^(2))+2x(x^(2)+1)(dy)/(dx)=

    A
    `4`
    B
    `2`
    C
    `1`
    D
    `0`

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