Question

If $y=x+e^x$, then what will be the value of $\frac{d^{2}x}{d{y}^2}$?

Answer 1

We have,

$y=x+e^x$

On differentiating both sides w.r.t $x$, we get

$\frac{dy}{dx}=1+e^x$

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

On differentiating both sides w.r.t $y$, we get

$\frac{d^{2}x}{d{y}^2}=\frac{(1+e^x)\times 0-1\times (0+e^x)}{{(1+e^x)}^2}\times \frac{dx}{dy}$

$\frac{d^{2}x}{d{y}^2}=\frac{-e^x}{{(1+e^x)}^2}\times \frac{1}{(1+e^x)}$

$\frac{d^{2}x}{d{y}^2}=\frac{-e^x}{{(1+e^x)}^3}$

Hence, this is the answer.