If-e-y-x-1-1-then-prove-that-d-2-y-dx-2-dy-dx-2-

Question Number 207122 by MATHEMATICSAM last updated on 07/May/24

If e^{y} (x + 1) = 1 then prove that

\frac{d^{2} y}{{d x}^{2}} = {(\frac{d y}{d x})}^{2} .

Answered by BaliramKumar last updated on 07/May/24

e^{y} = \frac{1}{x + 1}

y = - \ln (x + 1)

\frac{dy}{dx} = - \frac{1}{x + 1} \Rightarrow {(\frac{dy}{dx})}^{2} = {(\frac{1}{x + 1})}^{2}

\frac{dy}{dx} = - {(x + 1)}^{- 1}

\frac{d^{2} y}{{dx}^{2}} = - [- 1 {(x + 1)}^{- 2}]

\frac{d^{2} y}{{dx}^{2}} = {(x + 1)}^{- 2} = {(\frac{1}{x + 1})}^{2}

\frac{d^{2} y}{{dx}^{2}} = {(\frac{dy}{dx})}^{2}