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

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

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`x=e^(x/y)`
`ln x=x/y`
`y ln x=x`
Differentiate both sides
`y(1/x dx)+ln x dy=dx`
`ln xdy=frac{x-y}{x}dx`
`dy/dx=(x-y)/(xlnx)`
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
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  2. 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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  3. If x=e^(x/y),prove that (dy)/(dx)=(x-y)/(xlogx)

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

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