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If xsqrt(1+y)+ysqrt(1+x)=0, prove that (...

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

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Verified by Experts

`"We have "xsqrt(1+y)+ysqrt(1+x)=0`
`"or "xsqrt(1+y)=-ysqrt(1+x)`
`"or "x^(2)(1+y)=y^(2)(1+x)" [On squaring both sides]"`
`"or "x^(2)-y^(2)=y^(2)x-x^(2)y`
`"or "(x+y)(x-y)=-xy(x-y)`
`"or "x+y=-xy" "[because x- yne 0" as "x = y" does not satisfy the given equation"]`
`"or "y=-(x)/(1+x)`
`"or "(dy)/(dx)=-{((1+x)xx1-x(0+1))/((1+x^(2)))}=-(1)/((1+x^(2)))`
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RD SHARMA ENGLISH-DIFFERENTIATION-All Questions
  1. If x y=1 , prove that (dy)/(dx)+y^2=0 .

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

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

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

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

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

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  7. 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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  8. If y=xsin(a+y) , prove that (dy)/(dx)=(sin ^ 2(a+y))/(sin(a+y)-y\ co...

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

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

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  11. 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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  12. sin(xy) + (x)/(y) = x^(2) - y

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

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

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

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

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

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  18. Differentiate the following function with respect to x : x^x

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  19. Differentiate (sinx)^(logx) with respect to x :

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  20. Differentiate (cosx)^x with respect to x :

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