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If y=log(sqrt(x)+1/(sqrt(x))) , prove th...

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

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To solve the problem, we need to differentiate the function \( y = \log\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) \) and show that \( \frac{dy}{dx} = \frac{x - 1}{2x(x + 1)} \). ### Step-by-Step Solution 1. **Start with the function:** \[ y = \log\left(\sqrt{x} + \frac{1}{\sqrt{x}}\right) \] ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If y=sqrt(x+1)+sqrt(x-1) , prove that sqrt(x^2-1)(dy)/(dx)=1/2y

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

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

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  4. If y=logsqrt((1-tanx)/(1+tanx)) , prove that (dy)/(dx)=-sec2x .

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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