If `y=tan^(-1)x` , find `(d^2y)/(dx^2)` in terms of `y` alone.

Answer

N/a

Explanation

We have, `y=tan^(-1)x` [On differentiating w.r.t. x]
`:. (dy)/(dx) = (1)/(1+x^(2))` [again differentiating w.r.t.x]
Now, `(d^(2)y)/(dx^(2)) = (d)/(dx)(1+x^(2))^(-1)`
`= - 1(1+x^(2))^(-2).(d)/(dx)(1+x^(2))`
`= - (1)/((1+x^(2))^(2)).2x`
`= (-2tany)/((1+tan^(2)y)^(2))` , `[:' y = tan^(-1)xrArr tany=x]`
`=(-2tany)/((sec^(2)y)^(2))`
`=-2'(siny)/(cosy).cos^(2)y.cos^(2)y`
`=-sin^(2)y.cos^(2)y` , `[:'sin2x=2sinxcosx]`