Question:
If $y=x+e^{x}$, find $\frac{d^{2} x}{d y^{2}}$
Solution:
Given:
$y=x+e^{x}$
$\frac{\mathrm{d}^{2} \mathrm{x}}{\mathrm{d}^{2} \mathrm{y}}=\frac{1}{\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}}$
$\frac{d y}{d x}=1+e^{x}$
$\frac{d^{2} y}{d x^{2}}=e^{x}$
$\frac{d^{2} x}{d^{2} y}=\frac{1}{e^{x}}$
$=e^{-x}$