Derivative of ` tan^(-1)((x)/(sqrt( 1 - x^(2))))` with respect to
` sin^(-1) (3x - 4x^(3)) ` is

Options

A. `(1)/(sqrt(1-x^(2))`
B. `(3)/(sqrt(1-x^(2)))`
C. 3
D. `(1)/(3)`

Answer

D

Explanation

Let `u=tan^(-1) ((x)/(sqrt(1-x^(2))` and
`v=sin^(-1)(3x-4x^(3))`
put x= `sin theta`, then
`u=tan^(-1) ((sin theta)/(sqrt(1-sin^(2)theta))`
and `v=sin^(-1)(3 sin theta-4sin^3 theta)`
`Rightarrow u=tan^(-1) ((sin theta)/(cos theta))`
and `v=sin^(-1) (sin 3 theta)`
`Rightarrow u=tan^(-1) (tan theta)`
and `v=sin^(-1)(sin 3 theta)`
`Rightarrow u=theta and v=3 theta`
`Rightarrow u=sin^(-1) x and v=3 sin^(-1) x`.
On differentiating both sides w.r.t.x. we get
`(du)/(dx)=(1)/(sqrt(1-x^(2)) and (dv)/(dx)=3 xx (1)/(sqrt(1-x^(2))`
`therefore therefore (du)/(dv)=((du)/(dx))/((dv)/(dx))= ((1)/(sqrt(1-x^(2))/((3)/(sqrt(1-x^(2)))=(1)/(3)`