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Using mathematical induction prove that ...

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

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NCERT GUJARATI-CONTINUITY AND DIFFERENTIABILITY-MISCELLANEOUS EXERCISE ON CHAPTER 5
  1. sin^(-1)(xsqrt(x)),0 le x le 1

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  2. Differentiate w.r.t.x the function in Exercises 1 to 11. (cos^(-1)""...

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  3. cot^-1{sqrt(1+sinx)+sqrt(1-sinx)}/{sqrt(1+sinx)-sqrt(1-sinx)}

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  4. Differentiate w.r.t. x the function(logx)^(logx), x >1

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  5. Differentiate w.r.t.x the function in Exercises 1 to 11. cos( a cos ...

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  6. (sinx-cosx)^((sinx-cosx)),(pi)/(4) ltx lt (3pi)/(4)

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  7. Differentiate w.r.t.x the function in Exercises 1 to 11. x^(x)+x^(a)...

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  8. x^(x^(2)-3)+(x-3)^(x^2)," for "x gt 3.

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  9. Find (dy)/(dx),if y=12(1-cost), x=10(t-sint),-pi/2<t<pi/2

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  10. Find (dy)/(dx), if y=sin^(-1)x+sin^(-1)sqrt(1-x^2),-1lt=xlt=1.

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  11. xsqrt(1+y)+ysqrt(1+x)=0, then (dy)/(dx)=

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

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  13. If cosy=xcos(a+y) , with cosa!=+-1 , prove that (dy)/(dx)=(cos^2(a+y))...

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  14. If x=a(cos t+t\ s in\ t) and y=a\ (sint-t\ cos t),\ \ then find (d^2\...

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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) for all...

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  17. Using the fact that s in (A + B) = s in A cos B + cos A s in Band the ...

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

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

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

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