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

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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AI Generated Solution

To prove that \(\frac{dy}{dx} = -\frac{y(x^2y + x + y)}{x(xy^2 + x + y)}\) given the equation \(xy \log(x+y) = 1\), we will differentiate both sides of the equation with respect to \(x\) and simplify. ### Step 1: Differentiate both sides Start with the equation: \[ xy \log(x+y) = 1 \] Differentiating both sides with respect to \(x\): ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If sec((x+y)/(x-y))=a , prove that (dy)/(dx)=y/x .

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

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

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

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

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

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

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

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

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

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

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

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

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  17. Differentiate x^(sqrt(x)) with respect to x :

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

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

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  20. Differentiate x^cos^((-1)x) with respect to x :

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