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If y=x\ sin^(-1)x+sqrt(1-x^2) , prove th...

If `y=x\ sin^(-1)x+sqrt(1-x^2)` , prove that `(dy)/(dx)=sin^(-1)x`

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To prove that \(\frac{dy}{dx} = \sin^{-1} x\) for the function \(y = x \sin^{-1} x + \sqrt{1 - x^2}\), we will differentiate \(y\) step by step. ### Step-by-Step Solution: 1. **Write down the function**: \[ y = x \sin^{-1} x + \sqrt{1 - x^2} \] ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If y=e^xcosx , prove that (dy)/(dx)=sqrt(2)\ e^xcos(x+pi/4)

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  2. If y=1/2log((1-cos2x)/(1+cos2x)) , prove that (dy)/(dx)=2\ cos e c\ 2x

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  3. If y=x\ sin^(-1)x+sqrt(1-x^2) , prove that (dy)/(dx)=sin^(-1)x

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  4. If y=sqrt(x^2+a^2) , prove that y(dy)/(dx)-x=0

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  5. If y=e^x+e^(-x) , prove that (dy)/(dx)=sqrt(y^2-4)

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  6. If y=sqrt(a^2-x^2) , prove that y(dy)/(dx)+x=0

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  7. If x y=4 , prove that x((dy)/(dx)+y^2)=3y

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  8. Prove that d/(dx){x/2sqrt(a^2-x^2)+(a^2)/2sin^(-1)x/a}=sqrt(a^2-x^2)

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  9. Differentiate sin^(-1)(sinx),\ \ x in [0,\ 2pi]

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  10. Differentiate cos^(-1)(cosx),\ \ x in [0,\ 2pi]

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  11. Differentiate tan^(-1)(tanx),\ \ x in [0,\ pi]-{pi/2}

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  12. Differentiate sin^(-1)(2xsqrt(1-x^2)) with respect to x , if -1/(sqrt(...

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  13. Differentiate sin^(-1)(2xsqrt(1-x^2)) with respect to x , if 1/(sqrt(2...

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  14. Differentiate sin^(-1)(2xsqrt(1-x^2)) with respect to x theta in [-(pi...

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  15. Differentiate sin^(-1)(3x-4x^3) with respect to x , if -1/2ltxlt1/2

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  16. Differentiate sin^(-1)(3x-4x^3) with respect to x , if 1/2ltxlt1

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  17. Differentiate sin^(-1)(3x-4x^3) with respect to x , if -1ltxlt-1/2

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  18. Differentiate cos^(-1)(2x^2-1) with respect to x , if 0ltxlt1

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  19. Differentiate cos^(-1)(2x^2-1) with respect to x , if -1ltxlt0

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  20. Differentiate cos^(-1)(1-2x^2) with respect to x , if 0ltxlt1

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