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If x^(16)y^9=(x^2+y)^(17) , prove that x...

If `x^(16)y^9=(x^2+y)^(17)` , prove that `x(dy)/(dx)=2y`

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Given that,
`x^(16)y^9=(x^2+y)^(17)`
taking lo on boths sides we get,
`=>16logx+9logy=17log(x^2+y)`
taking derivative w.r.t. `x` on both sides we get,
`=>16/x+9/y*(dy)/(dx) =17 /(x^2+y) (2x+(dy)/(dx))`
`=>16/x-(34x)/(x^2+y)=(dy)/(dx)((-9x^2+8y)/y)`
`=>(dy)/(dx)=(2y)/x`
`=>x(dy)/(dx)=2y`
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RD SHARMA-DIFFERENTIATION-Solved Examples And Exercises
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  2. If x^(13)y^7=(x+y)^(20) , prove that (dy)/(dx)=y/x

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

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

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

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

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

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

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

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

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

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

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

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

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