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If x^y = e^(x-y) then prove that (dy)/(d...

If `x^y = e^(x-y)` then prove that `(dy)/(dx)=logx/(1+logx)^2`.

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SHARAM PUBLICATION-CONTINUITY AND DIFFERENTIABILITY-EXAMPLE
  1. If y=tan^(-1)((a)/(x))+logsqrt((x-a)/(x+a)), then prove that (dy)/(dx)...

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  2. If (tan^(-1)x)^y + y^(cotx) =1then find (dy)/(dx).

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  3. If x^y = e^(x-y) then prove that (dy)/(dx)=logx/(1+logx)^2.

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  4. If y =x log (x/(a+bx))then prove that x^3(d^2y)/(dx^2)=(x(dy)/(dx)-y)^...

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  7. Differentiate the following function w.r.t. x, (logx)^(x)+x^(logx).

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  8. If y=log[x+sqrt(x^(2)+1)], then prove that (x^(2)+1)(d^(2)y)/(dx^(2))+...

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  9. If x=acos^(3)theta,y=a sin^(3)theta,then find (d^2y)/dx^(2)

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  10. If sin y = x sin (a + y) then show that (dy)/(dx)=(sin^2(a+y))/(sina)

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  11. Find dy/dx if x^m y^n = (x/y)^(m+n)

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  12. Find (dy)/(dt), when y=sin^(-1)(2sqrt(t^(2)-1)/(t^(2)))

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  13. If x=asec theta, y=b tan theta, then prove that (d^(2)y)/(dx^(2))=-(b^...

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  14. IF x^y=e^(x-y) then show that (dy)/(dx)=(logx)/[log(xe)]^2.

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  15. If sin y = x sin (a + y) then show that (dy)/(dx)=(sin^2(a+y))/(sina)

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  16. If x=a(cost+tsint) and y=a(sint-tcost), then find (d^(2)y)/(dx^(2)) at...

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  17. If cosx=sqrt(1/(1+t^2)),sin y=(2t)/(1+t^2) then show that (dy)/(dx) is...

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  18. Find (dy)/(dx), when y^(x)=x^(siny)

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  19. If y^2 cot x = x^2 cot y then find (dy)/(dx).

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  20. Find the derivative of x^(sinx) w.r.t.x.

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