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Using mathematical induction prove that ...

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

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PRADEEP PUBLICATION-CONTINUITY AND DIFFERENTIABILITY-EXERCISE
  1. If x = a(cost + t sint) and y = a(sint - tcost), find (d^2y)/dx^2

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

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  3. Using mathematical induction prove that d(x^n)/dx = nx^(n-1) for all p...

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  4. Using the fact that sin(A+B) = sinA cosB + cosA sinB and the differen...

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

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  6. If y = |[f(x),g(x),h(x)],[1,m,n],[a,b,c]|, prove that dy/dx = |[f'(x)...

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  7. If y = e^(acos^-1x), -1lexle1, show that (1-x^2)(d^2y)/dx^2 - x(dy/dx)...

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  8. Fill in the blanks: The function f(x)= lfloor x rfloor is discontinu...

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  9. The number of points at which the function f(x)=(1)/(log|x|) is discon...

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  10. Fill in the blanks: An example of a function which is continuous eve...

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  11. If f(x)={(ax+1" if "xge1),(x+2" if "xlt1):} is continuous, then 'a' sh...

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  12. Find ainR so that f(x) = {:{(x^2, x ge0),(ax,x<0):} is continuous at 0...

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  13. Fill in the blanks: For all x in (6,7), d/(dx) ([x]) = ……………. .

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  14. Fill in the blanks: Derivative of sin x w.r.t cos x is …………… .

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  15. Fill in the blanks: Derivative of x^2 wr.t x^3 is …………… .

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  16. Fill in the blanks: If xge0, then d/(Dx) {cos^-1((1-x^2)/(1+x^2))} i...

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  17. The derivative of sin x cos x w.r.t. x is

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  18. Fill in the blanks: The derivative of log(10) x w.r.t x is …………….. ...

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  19. Fill in the blanks: If f(x) = |sin x|, then f'(pi/4) is equal to ………...

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  20. Fill in the blanks: Domain of continuity of the function f(x) =|x-1|...

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