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y=tan^(-1)""((sqrt(1+x^(2))-1)/(x)),x ne...

`y=tan^(-1)""((sqrt(1+x^(2))-1)/(x)),x ne 0`

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Verified by Experts

The correct Answer is:
`(1)/(2)((1)/(1+x^(2)))`
`"Let "y=tan^(-1)((sqrt(1+x^(2))-1)/(x))."Putting "x= tan theta," we get"`
`y=tan^(-1)((sec theta-1)/(tan theta))`
`=tan^(-1)((1-cos theta)/(sin theta))`
`=tan^(-1)((2sin^(2)""(theta)/(2))/(2 sin""(theta)/(2)cos""(theta)/(2)))`
`=tan^(-1)(tan""(theta)/(2))=(1)/(2)theta=(1)/(2)tan^(-1)x`
`therefore" "(dy)/(dx)=(1)/(2)((1)/(1+x^(2)))`
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CENGAGE-DIFFERENTIATION-Exercise 3.2
  1. Find (dy)/(dx) in the following : y= sin^(-1) (2x sqrt(1-x^(2))), (1...

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  2. Find (dy)/(dx) if y=tan^(-1)(4x)/(1+5x^2)+tan^(-1)(2+3x)/(3-2x)

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  3. y=tan^(-1)""((sqrt(1+x^(2))-1)/(x)),x ne 0

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  4. y=tan^(-1)(x/(1+sqrt(1-x^2)))

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  5. Find (dy)/(dx) for the function: y=sin^(-1)sqrt((1-x))+cos^(-1)sqrt(x)

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  6. Find (dy)/(dx) for the function: y=sqrt(sinsqrt(x))

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  7. Differentiate : y = e^(sin 2x)

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  8. Find (dy)/(dx) for the function: y=logsqrt(sinsqrt(e^x))

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  9. Find (dy)/(dx) for the function: y=a^((sin^(-1)x))^2

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  10. Find (dy)/(dx) if y=log{e^x((x-2)/(x+2))^(3/4)}

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  11. y=sin^(-1)[sqrt(x-a x)-sqrt(a-a x)] then prove that 1/(2sqrtx sqrt(1-x...

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  12. Find (dy)/(dx) for the functions: y=x^3e^xsinx

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  13. Find (dy)/(dx) for the function: y=(log)esqrt((1+sinx)/(1-sinx)),w h e...

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  14. Find (dy)/(dx) for the functions: y=(sinx)/(x+cosx)

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  15. If y=(1+x)(1+x^2)(1+x^4)......(1+x^(2n)), then find (dy)/(dx)a tx=0.

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  16. If xsqrt(1+y)+ysqrt(1+x)=0, prove that (dy)/(dx)=-1/((1+x)^2)

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  17. If g is the inverse function of fa n df^(prime)(x)=sinx ,t h e ng^(pri...

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  18. sin^(-1) ((2x)/(1 + x^(2)))

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  19. Find (dy)/(dx) in the following : y = tan^(-1)((3x-x^(3))/(1-3x^(2))...

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  20. Find (dy)/(dx) in the following : y= sec^(-1) ((1)/(2x^(2)-1)), 0 lt...

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