If ` y = tan ^-1[x/(1+sqrt(1-x^2))] "then find" dy/dx` – InterviewSolution

InterviewSolution

1.	If ` y = tan ^-1[x/(1+sqrt(1-x^2))] "then find" dy/dx`
Answer» Let `x=sin theta` ` therefore y=tan ^-1[(sin theta)/(1+sqrt(1-sintheta))]` `=tan^-1[(sin theta)/(1+cos theta)]=tan ^-1[(2sin ""theta/2cos ""theta/2)/(2cos^2""theta/2)]` `tan^-1(tan""theta/2)=theta/2=1/2sin^-1x` `rArr dy/dx=1/2d/dxsin^-1x=1/(2sqrt(1-x^2))`

Discussion

No Comment Found

Related InterviewSolutions

The number of points of discontinuity of `f(x)=[2x]^(2)-{2x}^(2)` (where [ ] denotes the greatest integer function and { } is fractional part of x) in the interval `(-2,2)`, isA. 1B. 6C. 2D. 5
The function `f(x)=(x^3)/8-s inpix+4in[-4,4]`does not take the value`-4`b. `10`c. `18`d. `12`A. `-4`B. 10C. 18D. 12
Let `f(x)={{:(-3+|x|",",-ooltxlt1),(a+|2-x|",",1lexltoo):}` and `g(x)={{:(2-|-x|",",-ooltxlt2),(-b+"sgn(x),",2lexltoo):}` where sgn(x) denotes signum function of x. If `h(x)=f(x)+g(x)` is discontinuous at exactly one point, then which of the following is not possible?A. `a=-3,b=0`B. `a=0,b=1`C. `a=2,b=1`D. `a=-3,b=1`
If `f(x) = sgn(x^5)`, then which of the following is/are false (where sgn denotes signum function)A. continuous and differentiableB. continuous but not differentiableC. differentiable but not continuousD. neither continuous nor differentiable
If x = `a cos^(3) theta, y = a sin ^(3) theta` then `sqrt(1+((dy)/(dx))^(2))`=?A. `tan^(2) theta`B. `sec ^(2) theta`C. `sec theta`D. `|sec theta|`
If `y=[x+sqrt(x^2+a^2)]^n` then prove that `(dy)/dx=(ny)/sqrt(x^2+a^2)`
`x=2 cos^2 t, y= 6 sin ^2 t `
`x=sqrt(sin 2t),y=sqrt(cos 2 t)`
If `x=sin^-1""(2t)/(1+t^2)` and `y=tan ^-1""(2t)/(1-t) " the n find " dy/dx.`
`x = "sin" t, y = "cos" 2t`