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If y=(xsin^(-1)x)/(sqrt(1-x^2)) , prove ...

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

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AI Generated Solution

To prove that \((1-x^2)\frac{dy}{dx} = x + \frac{y}{x}\) given that \(y = \frac{x \sin^{-1} x}{\sqrt{1-x^2}}\), we will follow these steps: ### Step 1: Differentiate \(y\) with respect to \(x\) Given: \[ y = \frac{x \sin^{-1} x}{\sqrt{1-x^2}} \] ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If y=logsqrt((1-tanx)/(1+tanx)) , prove that (dy)/(dx)=-sec2x .

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

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

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  4. If y=(e^x-e^(-x))/(e^x+e^(-x)) , prove that (dy)/(dx)=1-y^2

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  5. If y=(x-1)log(x-1)-(x+1)log(x+1) , prove that (dy)/(dx)=log((x-1)/(1+x...

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  6. If y=e^xcosx , prove that (dy)/(dx)=sqrt(2)\ e^xcos(x+pi/4)

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

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

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

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

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

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

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

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

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

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

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

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

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

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