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If x y=1 , prove that (dy)/(dx)+y^2=0 ....

If `x y=1` , prove that `(dy)/(dx)+y^2=0` .

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Given as `x y=1`

Differentiating with respect to `x`, $$ \begin{aligned} &\frac{d}{d x}(x y)=\frac{d}{d x}(1) \\ &\Rightarrow x \frac{d y}{d x}+y \frac{d}{d x}(x)=0 \end{aligned} ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If sqrt(1-x^2) + sqrt(1-y^2)=a(x-y), prove that (dy)/(dx)= sqrt((1-y^2...

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  2. If ysqrt(1-x^2)+xsqrt(1-y^2)=1 show that (dy)/(dx)=-sqrt((1-y^2)/(1-x...

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  3. If x y=1 , prove that (dy)/(dx)+y^2=0 .

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  4. If x y^2=1 , prove that 2(dy)/(dx)+y^3=0

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  5. If xsqrt(1+y)+ysqrt(1+x)=0, prove that (dy)/(dx)=-1/((x+1)^2)

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  6. If logsqrt(x^2+y^2)=tan^-1(y/x),then dy/dx is

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  7. If sec((x+y)/(x-y))=a , prove that (dy)/(dx)=y/x .

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  8. If tan^(-1)((x^2-y^2)/(x^2+y^2))=a , prove that (dy)/(dx)=x/y((1-tana)...

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  9. If x y\ log(x+y)=1, prove that (dy)/(dx)=-(y(x^2y+x+y))/(x(x y^2+x+y))...

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  10. If y=xsin(a+y) , prove that (dy)/(dx)=(s in^2(a+y))/(sin(a+y)-y\ cos\ ...

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  11. If xsin(a+y)+sinacos(a+y)=0 , prove that (dy)/(dx)=(s in^2(a+y))/(sina...

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

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

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  14. If sin(x y)+y/x=x^2-y^2 , find (dy)/(dx) .

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  15. If tan(x+y)+tan(x-y)=1 , find (dy)/(dx) .

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  16. If e^x+e^y=e^(x+y) , prove that (dy)/(dx)=-(e^x(e^y-1))/(e^y(e^x-1)) o...

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

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

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  19. If sqrt(y+x) +sqrt(y-x) =c show that dy/dx = y/x -sqrt((y^2/x^2)-1)

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  20. Differentiate the following functions with respect to x : (I)x^x (ii...

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