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If x^y* y^x=1 , prove that (dy)/(dx)=-(y...

If `x^y* y^x=1` , prove that `(dy)/(dx)=-(y(y+xlogy))/(x(ylogx+x))`

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To prove that \(\frac{dy}{dx} = -\frac{y(y + x \log y)}{x(y \log x + x)}\) given the equation \(x^y \cdot y^x = 1\), we can follow these steps: ### Step 1: Take the logarithm of both sides Starting with the equation: \[ x^y \cdot y^x = 1 \] Taking the natural logarithm on both sides gives: ...
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
  1. If y=sin(x^x) , prove that (dy)/(dx)=cos(x^x) .x^x(1+logx)

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  2. If x^x+y^x=1 , prove that (dy)/(dx)=-{(x^x(1+logx)+y^x. logy)/(x . y^(...

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  3. If x^y* y^x=1 , prove that (dy)/(dx)=-(y(y+xlogy))/(x(ylogx+x))

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  4. If y^x+x^y=(x+y)^(x+y) find dy/dx

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  5. If x^m\ y^n=1 , prove that (dy)/(dx)=-(m y)/(n x)

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

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  7. If (sinx)^y=(cosy)^x , prove that (dy)/(dx)=(logcosy-ycotx)/(logsinx+x...

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  8. If (cosx)^y=(tany)^x , prove that (dy)/(dx)=(logtany+ytanx)/(logcosx-x...

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  9. If e^x+e^y=e^(x+y) , prove that (dy)/(dx)+e^(y-x)=0

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

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

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

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

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  14. If (sinx)^y=x+y , prove that (dy)/(dx)=(1-(x+y)ycotx)/((x+y)logsinx-1)

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  15. If (x-y) \ e^(x/(x-y))=a , prove that y(dy)/(dx)+x=2y .

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

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  17. Find the derivative of the function f(x) given by f(x)=(1+x)(1+x^2)...

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  18. If y=log(x^2+x+1)/(x^2-x+1)+2/(sqrt(3))t a n^(-1)((sqrt(3)x)/(1-x^2)),...

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  19. If y=(sinx-cosx)^(sinx-cosx) ,pi/4 ltxlt (3pi)/ 4 find dy/dx

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  20. If x y=e^(x-y) , find (dy)/(dx) .

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