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If u = tan^(-1)"" (sqrt ( 1+x^2)-1)/(x) ...

If `u = tan^(-1)"" (sqrt ( 1+x^2)-1)/(x) and v = tan^(-1) x`, find `(du)/(dv)`

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The correct Answer is:
`(dv)/(d theta) = 1 (du)/(dv) = (du// d theta)/( dv // d theta) = (1//2)/( 1) = (1)/(2)`
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FULL MARKS-DIFFERENTIAL CALCULUS - DIFFERENTIABILITY AND METHODS OF DIFFERENTIATION-EXERCISE 4 (Differentiate the following)
  1. If cos ( xy) = x, show that (dy)/(dx) = -((1+ y sin (xy)))/(x sin xy)

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  2. Differentiate the following w.r.t. x; tan^(-1) sqrt((1 - cos x)/(1 + c...

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  3. Find the derivatives of the following : tan^(-1) ((6x)/(1-9x^2))

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  4. Find the derivatives of the following : cos [2 tan^(-1) sqrt((1 - x)/(...

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  5. Find the derivatives of the following : x = a cos^(3) t , y = a sin^(3...

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  6. If x =a ( cos t + t sin t), y = a [sin t-t cos t] then find dy/dx.

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  7. Find the derivatives of the following : x = (1-t^2)/(1+t^2), y = (2t)...

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  8. Find the derivatives of the following : cos^(-1) ((1 -x^2)/(1+x^2))

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  9. Find the derivatives of the following : sin^(-1) (3x-4x^3)

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  10. Find the derivatives of the following : tan^(-1) ((cos x + sin x)/(cos...

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  11. Find the derivative of sin x^2 with respect to x^2.

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  12. Find the derivative of sin^(-1) ((2x)/(1+x^2)) with respect to tan^(-1...

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  13. If u = tan^(-1)"" (sqrt ( 1+x^2)-1)/(x) and v = tan^(-1) x, find (du)/...

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  14. Find the derivative with tan^(-1) ((sinx )/(1 + cos x)) with respect t...

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  15. If y = sin^(-1) x then find y''.

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  16. If y = e^(tan^(-1) x). find dy/dx

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  17. If y = (sin^(-1) x)/(sqrt(1-x^2)) show that (1-x^2) y2 - 3xy1 - y =0

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  18. If x= a (theta+ sin theta), y =a ( 1- cos theta) then prove that at th...

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  19. If cos y = x cos (a+y) Then prove that (dy)/(dx) = (cos^(2) (a+y))/(si...

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  20. If y = ( cos^(-1) x)^2, prove that (1-x^2) (d^2 y)/(dx^2) - x(dy)/(dx)...

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