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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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BANSAL-LIMITS AND DERIVATIVES-All Questions
  1. If y=e^acos^((-1)x) , then prove that (1-x^2)y2-x y1-a^2y=0 .

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  2. If (x-a)^2+(y-b)^2=c^2\ \ (c >0) then ([1+((dy)/(dx))^2]^(3//2))/((d^2...

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

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  4. If y=e^tan^((-1)x) , prove that (1+x^2)y2+(2x-1)y1=0 .

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  5. If y=e^(asin^(-1)x) then show that (1-x^2)y(n+2)-(2n+1)xy(n+1)-(n^2+a^...

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  6. Find (dy)/(dx), if x=acostheta , y=asintheta

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  7. Find (dy)/(dx) , if x=a t^2 , y=2a t

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  8. Find (dy)/(dx) , if x^(2/3) +y^(2/3)=a^(2/3)

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  9. Find (dy)/(dx) , if x=2a t^2 , y=a t^4

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  10. Find (dy)/(dx) , if x=(sin^3t)/(sqrt(cos2t)) , y=(cos^3t)/(sqrt(cos2t)...

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  11. Find (dy)/(dx) , if x=acostheta , y=bcostheta

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  12. Find (dy)/(dx) , if x=costheta-cos2theta , y=sintheta-sin2theta

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  13. Find (dy)/(dx), when x=a e^(theta)(sintheta-costheta),y=a e^(theta)(si...

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  14. Find (dy)/(dx) , if x=a(cost+logtant/2) , y=asint

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  15. Find (dy)/(dx) , if x=asectheta , y=btantheta

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  16. Find (dy)/(dx) , if x=a(costheta+thetasintheta) , y=a(sintheta-thetaco...

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  17. If x=sqrt(a^sin^((-1)t)) , y=sqrt(a^cos^((-1)t)) , show that (dy)/(...

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  18. If y = x^3 + tan x, show that (d^2y)/dx^2= 6x + 2 sec^2 tanx.

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  19. If y=3e^(2x)+2e^(3x) , prove that (d^2y)/(dx^2)-5(dy)/(dx)+6y=0 .

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

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