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

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

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We have , `y ^x =e ^(y−x)`
Taking log both side `=>logy ^x ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If x^y=y^x ,then find (dy)/(dx)

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  2. If (sin"x")^y=(cos"y")^x ,p rov et h a t(dy)/(dx)=(logcosy-ycotx)/(log...

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

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  4. If y=(a x^2)/((x-a)(x-b)(x-c))+(b x)/((x-b)(x-c))+c/(x-c)+1 find dy/dx

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  5. If x=e^(x/y),prove that (dy)/(dx)=(x-y)/(xlogx)

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  6. If y=xsiny ,prove that (dy)/(dx)=y/(x(1-xcosy)

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  7. If x ylog(x+y)=1,p rov et h a t(dy)/(dx)=-(y(x^2y+x+y))/(x(x y^2+x+y))

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  8. If (sin"x")^y=x+y ,p rov et h a t(dy)/(dx)=(1-(x+y)ycotx)/((x+y)logsin...

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  9. Differentiate: (logx)^x+x^(logx) w i t h respect to x

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  10. Find the derivative of (sqrt(x)(x+4)^(3/2))/((4x-3)^(4/3)) with respec...

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  11. Differentiate the following functions with respect to x : (cos"x")^x ...

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  12. Differentiate the following functions with respect to x : (i) (x^x...

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  13. If y={(log)(cosx)sinx}{(log)(sinx)cosx}^(-1)+sin^(-1)((2x)/(1+x^2)),f...

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  14. If sqrt(y+x)+sqrt(y-x)=c ,s howt h a t(dy)/(dx)=y/x-sqrt((y^2)/(x^2)-1...

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  15. If y=cos^(-1)sqrt((cos3x)/(cos^3x)),t h e ns howt h a t(dy)/(dx)=sqrt...

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  16. If sqrt(1-x^2)+sqrt(1-y^2)=a(x-y),p rov et h a t(dy)/(dx)=sqrt((1-y^2...

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  17. if x^2+y^2 = t - 1/t and x^4 + y^4 = t^2 + 1/t^2 then prove that dy/dx...

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  18. If y=btan^(-1)(x/a+tan^(-1)y/x),find(dy)/(dx)dot

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  19. Differentiate sin^(-1)((2x)/(1+x^2))with respect tocos^(-1)((1-x^2)/(1...

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  20. Differentiate sin^(-1)((2x)/(1+x^2))+cos^(-1)((1-x^2)/(1+x^2)) with re...

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