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If e^y(x+1)=1, prove that (d^2y)/(dx^2)=...

If `e^y(x+1)=1`, prove that `(d^2y)/(dx^2)=((dy)/(dx))^2`

Text Solution

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Given relation is `e^(y)(x+1)=1.`
Differentiating both sides w.r.t. x, we get
`(x+1)e^(y)(dy)/(dx)+e^(y).1=0`
`therefore" "(dy)/(dx)=-(1)/(x+1)`
Differentiating again w.r.t. x both sides, we get
`(d)/(dx)((dy)/(dx))=(d)/(dx)(-(1)/(x+1))`
`therefore" "(d^(2)y)/(dx^(2))=(1)/((x+1)^(2))`
`"From (1) and (2), we get "(d^(2)y)/(dx^(2))=((dy)/(dx))^(2).`
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