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Derivative of ` tan^(-1)((x)/(sqrt( 1 - x^(2))))` with respect to ` sin^(-1) (3x - 4x^(3)) ` isA. `(1)/(sqrt(1-x^(2))`B. `(3)/(sqrt(1-x^(2)))`C. 3D. `(1)/(3)` |
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Answer» Correct Answer - D 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)` |
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