Differentiate the following functions with respect to `x` : (i) `tan^(-1)((a+x)/(1-a x))` (ii) `tan^(-1)((acosx-bsinx)/(bcosx+asinx)),\ -pi/2 < x < pi/2 and a/b tanx > -1`

Explanation

`(i)`Given that,
`y=tan^(-1)((a+x)/(1-a x))`
`=>y=tan^-1 a+tan^-1 x`
`(dy)/(dx)=d/(dx) (tan^-1 a)+d/(dx) (tan^-1 x)`
`=0+1/(1+x^2)`
`(dy)/(dx)=1/(1+x^2)`


`(ii)` given that,
`y=tan^(-1)((acosx-bsinx)/(bcosx+asinx))`
`=>y=tan^-1[(a/b-tanx)/(1+a/b*tanx)]`
`=>y=tan^-1(a/b)-tan^-1(tanx)`
`=tan^-1(a/b)-x`
`(dy)/(dx)=-1`