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If (x-a)^2 + (y-b)^2 = c^2 , for some c ...

If `(x-a)^2 + (y-b)^2 = c^2` , for some `c > 0`, prove that `[1+(dy/dx)^2]^(3/2)/((d^2y)/dx^2)` is a constant independent of a and b.

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PRADEEP PUBLICATION-CONTINUITY AND DIFFERENTIABILITY-EXERCISE
  1. y = e^(m sin^(-1)x), prove that (1 - x^2) y2 - xy1 = m^2y

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  2. If x^2/a^2 - y^2/b^2 = 1, prove that ((d^2y)/dx^2) = (-b^4)/(a^2y^3)

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  3. If (x-a)^2 + (y-b)^2 = c^2 , for some c > 0, prove that [1+(dy/dx)^2]...

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  4. Find (d^2y)/dx^2, if x = a (theta - sin theta), y = a(1+cos theta)

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  5. If x = 2 cos theta - cos 2 theta, y = 2 sin theta - sin 2 theta, find ...

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  6. If x = a { cos t + log|tan t/2|} and y = a sin t, 0 < t < pi/2, find (...

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  7. If x=acos theta+bsintheta and y=a sintheta-bcostheta, then prove that ...

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  8. If y = x log (x/(a+bx)), prove that x^3 y 2 = (xy1 - y)^2

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  9. If x = g(t) and y = f(t), find (d^2y)/(dx^2) as a function of t.

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  10. If f(x) = |x|^3 show that f''(x) exists for all real x and find it.

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  11. Find the second derivative of the following functions: ax^3 + bx^2 +...

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  12. Find the second derivative of the following functions: a^x

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  13. Find the second derivative of the following functions: x sin x

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  14. Find the second derivative of the following functions: log (logx)

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  15. Find the second derivative of the following functions: e^(sinx)

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  16. Find the second derivative of the following functions: log(x+sqrt(x^...

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  17. Find the second derivative of the following functions: sin 3 x sin 6...

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  18. Find the second derivative of the following functions: x/(x^2 - 3x +...

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  19. If y = sin^(-1)x, then prove that (d^(2)y)/(dx^(2)) = (x)/((1 - x^(2...

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  20. If y = sin^-1x, then show that (1-x^2)(d^2y)/(dx^2)- x dy/dx = 0

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