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Using mathematical induction prove that `(d)/(dx) (x^(n))= n x^(n-1)` for all positive integers n.

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KUMAR PRAKASHAN-CONTINUITY AND DIFFERENTIABILITY-Miscellaneous Exercise - 5
  1. Differentiate w.r.t x the function sin^(-1) (x sqrtx), where 0 le x le...

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  2. Differentiate w.r.t x the function ("cos"^(-1) (x)/(2))/(sqrt(2x + 7))...

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  3. Differentiate w.r.t x the function 0 lt x lt (pi)/(2), cot^(-1) [(sqrt...

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  4. Differentiate w.r.t x the function (log x)^(log x), x gt 1

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  5. Differentiate w.r.t x the function cos (a cos x + b sin x), for some c...

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  6. Differentiate w.r.t x the function (pi)/(4) lt x lt (3pi)/(4), (sin x ...

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  7. Differentiate w.r.t x the function x^(x) + x^(a) + a^(x) + a^(a), for ...

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  8. Differentiate w.r.t x the function x^(x^(2)-3) + (x-3)^(x^(2)), for x ...

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  9. Find (dy)/(dx), if y= 12(1- cos t), x= 10 (t- sin t), -(pi)/(2) lt t l...

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  10. Find (dy)/(dx), If y= sin^(-1) x + sin^(-1) sqrt(1 -x^(2)), 0 le x le ...

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  11. If x sqrt(1 + y) + ysqrt(1 + x)= 0, for -1 lt x lt 1, prove that (dy)/...

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  12. If (x-a)^(2) + (y-b)^(2) = c^(2), for some c gt 0, prove that ([1 + ((...

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  13. If cos y= x cos (a + y), with cos a ne +- 1, prove that (dy)/(dx) = (c...

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  14. If x=a (cos t+ t sin t) and y= a (sin t - t cos t), find (d^(2)y)/(dx^...

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

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  16. Using mathematical induction prove that (d)/(dx) (x^(n))= n x^(n-1) fo...

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  17. Using the fact that sin (A +B) = sin A.cos B+ cos A sin B and the diff...

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  18. Does there exist a function which is continuous everywhere but not dif...

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  19. If y= |(f(x),g(x),h(x)),(l,m,n),(a,b,c)|, prove that (dy)/(dx)= |(f'(x...

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  20. If y = e^(a cos^(-1)x) show that (1-x^(2)) (d^(2)y)/(dx^(2))-x (dy)/(d...

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