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If y=sqrt(x+sqrt(x+sqrt(x+...tooo))), pr...

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

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To solve the problem, we start with the given equation: \[ y = \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}} \] ### Step 1: Rewrite the equation Since the expression inside the square root is the same as \( y \), we can rewrite the equation as: \[ y = \sqrt{x + y} \] ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If y=sqrt(x+sqrt(x+sqrt(x+...tooo))), prove that (dy)/(dx)=1/(2y-1)

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  2. Differentiate the following function with respect to x: x^(sin^(-1)...

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  3. Differentiate sin^(-1)xwith respect to cos^(-1)sqrt(1-x^2),if -1ltaxlt...

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  4. Differentiate the following functions with respect to x e^(sinsqrt(...

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  5. Differentiate the following functions with respect to x sin(logx)

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  6. Differentiate the following function sin^2(2x+1)

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  7. Differentiate the following function from first principles: sin^(-1...

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  8. Differentiate the following functions with respect to x : sin(3x...

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  9. Differentiate the following functions with respect to tan(x^0+45^0)

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  10. Differentiate the following function with respect to x : cos^(-1...

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  11. Differentiate the following functions with respect to tan5x^@

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  12. Differentiate the following functions with respect tox: (log)7(logx)

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  13. Differentiate the following function with respect to x: (i)e^(xl...

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  14. Differentiate the following function with respect to x: x^(cos^-...

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  15. Differentiate the following function with respect to x: (logx)^(co...

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  16. Differentiate the following function with respect to x:x^(1/x)

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

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

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

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  20. 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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